MixedElasticity2D.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "77d1baa2",
   "metadata": {},
   "source": [
    "### In this notebook we solve 2D Mixed Elasticity with assmbled/matrix-free approaches. Also inf-sup constant is computed for continuous and discontinuous pressure"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "5a2dadfd",
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots\n",
    "using LinearAlgebra\n",
    "using SparseArrays\n",
    "using FastGaussQuadrature\n",
    "using NLsolve\n",
    "using Random\n",
    "using DataFrames\n",
    "using Test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "c1b37930",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "my_spy (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "default(linewidth=3)\n",
    "function my_spy(A)\n",
    "    cmax = norm(vec(A), Inf)\n",
    "    s = max(1, ceil(30 / size(A, 1)))\n",
    "    spy(A, marker=(:square, s), c=:diverging_rainbow_bgymr_45_85_c67_n256, clims=(-cmax, cmax))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "ba69e57e",
   "metadata": {},
   "outputs": [],
   "source": [
    "function vander_legendre_deriv(x, k=nothing)\n",
    "    if isnothing(k)\n",
    "        k = length(x) # Square matrix by default\n",
    "    end\n",
    "    m = length(x)\n",
    "    Q = ones(m, k)\n",
    "    dQ = zeros(m, k)\n",
    "    Q[:, 2] = x\n",
    "    dQ[:, 2] .= 1\n",
    "    for n in 1:k-2\n",
    "        Q[:, n+2] = ((2*n + 1) * x .* Q[:, n+1] - n * Q[:, n]) / (n + 1)\n",
    "        dQ[:, n+2] = (2*n + 1) * Q[:,n+1] + dQ[:,n]\n",
    "    end\n",
    "    Q, dQ\n",
    "end\n",
    "\n",
    "function febasis1D(P, Q, Qmode)\n",
    "    x, _ = gausslobatto(P)\n",
    "    if Qmode == \"GAUSS\"\n",
    "        q, w = gausslegendre(Q)\n",
    "    elseif Qmode == \"GLL\"\n",
    "        q, w = gausslobatto(Q)\n",
    "    else\n",
    "        error(\"Qmode error! Choose GAUSS or GLL Quadrature points!\");\n",
    "    end\n",
    "    V, _ = vander_legendre_deriv(x)\n",
    "    Bp, Dp = vander_legendre_deriv(q, P)\n",
    "    B = Bp / V\n",
    "    D = Dp / V\n",
    "    x, q, w, B, D\n",
    "end\n",
    "\n",
    "meshgrid(x, y) = (repeat(x, outer=length(y)), repeat(y, inner=length(x)))\n",
    "\n",
    "function febasis2D(P, Q, num_comp, Qmode)\n",
    "    # coordinate, quadrature, and basis on reference element [-1,1]\n",
    "    x1_ref, q1_ref, w1_ref, B1, D1 = febasis1D(P, Q, Qmode)\n",
    "    \n",
    "    x, y = meshgrid(x1_ref, x1_ref)\n",
    "    x2_ref = [x;y]\n",
    "    qx, qy = meshgrid(q1_ref, q1_ref)\n",
    "    q2_ref = [qx;qy]\n",
    "    w2_ref = kron(w1_ref, w1_ref)\n",
    "    \n",
    "    B2 = kron(I(num_comp),kron(B1,B1))\n",
    "    Dx = kron(B1, D1)\n",
    "    Dy = kron(D1, B1)\n",
    "    # Grad for num_comp = 1\n",
    "    Dx = kron([1, 0], Dx)\n",
    "    Dy = kron([0, 1], Dy)\n",
    "    D2 = Dx + Dy\n",
    "    D2 = kron(I(num_comp),D2)\n",
    "    \n",
    "    x2_ref, q2_ref, w2_ref, B2, D2\n",
    "end\n",
    "\n",
    "struct FEBasis\n",
    "    P::Int\n",
    "    Pp::Int\n",
    "    Q::Int\n",
    "    num_comp::Int \n",
    "    Qmode::String\n",
    "    x_ref::Vector  # nodes at 2D ref element\n",
    "    q_ref::Vector  # quad pts for 2D ref element\n",
    "    w_ref::Vector  # weights for 2D ref element\n",
    "    B::Matrix  # Interpolation for u\n",
    "    D::Matrix  # Derivative for u\n",
    "    Bp::Matrix\n",
    "    p_is_tensor::Bool\n",
    "    function FEBasis(P, Pp, Q, num_comp, Qmode, p_is_tensor)\n",
    "        x_ref, q_ref, w_ref, B, D = febasis2D(P, Q, num_comp, Qmode)\n",
    "        if !p_is_tensor && Pp == 1 # This is constant P0 discontinuous element\n",
    "            Bp = ones(Q^2,1)\n",
    "        elseif !p_is_tensor && Pp == 3 # This is linear P1 discontinuous element\n",
    "            q_ref2 = reshape(q_ref, Q^2,:);\n",
    "            Bp = hcat(ones(Q^2), q_ref2[:,1], q_ref2[:,2])\n",
    "        elseif !p_is_tensor && Pp == 6 # This is quadratic P2 discontinuous element\n",
    "            q_ref2 = reshape(q_ref, Q^2,:);\n",
    "            Bp = hcat(ones(Q^2), q_ref2[:,1], q_ref2[:,2], q_ref2[:,1] .* q_ref2[:,2], \n",
    "                      q_ref2[:,1] .* q_ref2[:,1], q_ref2[:,2] .* q_ref2[:,2])\n",
    "        elseif p_is_tensor && Pp > 1 # This is Q element\n",
    "            _, _, _, Bp, _ = febasis2D(Pp, Q, 1, Qmode)\n",
    "        else\n",
    "            error(\"Unsupported dof for pressure field\");\n",
    "        end\n",
    "        \n",
    "        new(P, Pp, Q, num_comp, Qmode, x_ref, q_ref, w_ref, B, D, Bp, p_is_tensor)\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "fd50aecc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "P1_basis_phy (generic function with 1 method)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#define P1 basis on physical element\n",
    "function P1_basis_phy(q, Q)\n",
    "    # note q is quadrature pts in elmenet E obtained by q = Bx * Coord_E\n",
    "    qq = reshape(q, Q^2,:);\n",
    "    Bp = hcat(ones(Q^2), qq[:,1], qq[:,2])\n",
    "    return Bp\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "95e08806",
   "metadata": {},
   "source": [
    "# Assembled approach\n",
    "$\\sigma = -p I + 2\\mu \\epsilon_d $, $p=-\\kappa \\nabla \\cdot u $\n",
    "$$ \\int_{\\Omega} \\nabla v \\sigma= \\int_{\\Omega} v f, \\forall v$$\n",
    "$$ \\int_{\\Omega} q \\left(-\\nabla\\cdot u - p/\\kappa \\right)= 0, \\forall q$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "1cc6e43f",
   "metadata": {},
   "outputs": [],
   "source": [
    "function GetConnectivity(P, nx, ny)\n",
    "    \"\"\"This function returns the connectivity array\n",
    "\n",
    "    4-------5--------6\n",
    "    |       |        |\n",
    "    |       |        |\n",
    "    |       |        |\n",
    "    1-------2--------3\n",
    "    local numbering of one element is\n",
    "\n",
    "    3-------4\n",
    "    |       |\n",
    "    |       |\n",
    "    |       |\n",
    "    1-------2\n",
    "\n",
    "\n",
    "    Input:\n",
    "    ------\n",
    "    nx: number of element in x direction start from 1 NOT 0\n",
    "    ny: number of element in y direction start from 1 NOT 0\n",
    "    Output:\n",
    "    ------\n",
    "    idx: connectivity index of size (num_elem x P*P) based on above node numbering\n",
    "    idx_bc: index for 4 faces to apply bc: [bottom, right, top, left]\n",
    "    \"\"\"\n",
    "    n_elems = [nx, ny]\n",
    "    nodes_u = zeros(Int64, 2)\n",
    "    nodes_x = zeros(Int64, 2)\n",
    "    for d=1:2\n",
    "        nodes_u[d] = n_elems[d]*(P-1) + 1\n",
    "        nodes_x[d] = n_elems[d]*(2-1) + 1\n",
    "    end\n",
    "\n",
    "    num_elem = n_elems[1] * n_elems[2]\n",
    "    idx_x = zeros(Int64, 4, num_elem) # Coordinate connectivity\n",
    "    idx_u = zeros(Int64, P * P, num_elem) # Solution connectivity\n",
    "    for i=1:n_elems[1]\n",
    "        for j=1:n_elems[2]\n",
    "            ele = (j-1)*n_elems[1] + i\n",
    "            for ii=1:P\n",
    "                for jj=1:P\n",
    "                    if (ii < 3) && (jj < 3)\n",
    "                        idx_x[(ii-1) * 2 + jj, ele] = ((j-1) * (2 - 1) + ii-1) * nodes_x[1] + (i-1) * (2 - 1) + jj\n",
    "                    end\n",
    "                    idx_u[(ii-1) * P + jj, ele] = ((j-1) * (P - 1) + ii-1) * nodes_u[1] + (i-1) * (P - 1) + jj\n",
    "                end\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    idx_b = Int64[] # bottom face\n",
    "    idx_t = Int64[] # top face\n",
    "    for i=1:n_elems[1]\n",
    "        if i==1\n",
    "            append!(idx_b, idx_u[1:P,i])\n",
    "            append!(idx_t, idx_u[P*(P-1)+1:P*P,i + n_elems[1]*(n_elems[2] -1)])\n",
    "        else\n",
    "            append!(idx_b, idx_u[2:P,i])\n",
    "            append!(idx_t, idx_u[P*(P-1)+2:P*P,i + n_elems[1]*(n_elems[2] -1)])\n",
    "        end\n",
    "    end\n",
    "\n",
    "    idx_l = Int64[] # left face\n",
    "    idx_r = Int64[] # right face\n",
    "    for i=1:n_elems[1]:num_elem\n",
    "        if i==1\n",
    "            append!(idx_l, idx_u[1:P:P*(P-1)+1,i])\n",
    "            append!(idx_r, idx_u[P:P:P*P,i+n_elems[1]-1])\n",
    "        else\n",
    "            append!(idx_l, idx_u[P+1:P:P*(P-1)+1,i])\n",
    "            append!(idx_r, idx_u[2*P:P:P*P,i+n_elems[1]-1])\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    return idx_x, idx_u, idx_b, idx_r, idx_t, idx_l, nodes_u\n",
    "end\n",
    "\n",
    "function GetConnectivityPressure(Pp, nx, ny, p_is_continuous, p_is_tensor)\n",
    "    \"\"\"This function returns the connectivity array for pressure field\n",
    "    \"\"\"\n",
    "    n_elems = [nx, ny]\n",
    "    num_elem = n_elems[1] * n_elems[2]\n",
    "    \n",
    "    if !p_is_tensor && !p_is_continuous && Pp == 1\n",
    "        elem_dof_p = 1 \n",
    "        idx_p = zeros(Int64, elem_dof_p, num_elem) # Solution connectivity for p\n",
    "        for i=1:num_elem\n",
    "            idx_p[1, i] = i\n",
    "        end\n",
    "    elseif !p_is_tensor && !p_is_continuous && Pp == 3\n",
    "        elem_dof_p = 3\n",
    "        idx_p = zeros(Int64, elem_dof_p, num_elem) # Solution connectivity for p\n",
    "        k = 0\n",
    "        for i=1:num_elem\n",
    "            for j=1:elem_dof_p\n",
    "                idx_p[j, i] = j + k\n",
    "            end\n",
    "            k += elem_dof_p\n",
    "        end\n",
    "    elseif !p_is_tensor && !p_is_continuous && Pp == 6\n",
    "        elem_dof_p = 6\n",
    "        idx_p = zeros(Int64, elem_dof_p, num_elem) # Solution connectivity for p\n",
    "        k = 0\n",
    "        for i=1:num_elem\n",
    "            for j=1:elem_dof_p\n",
    "                idx_p[j, i] = j + k\n",
    "            end\n",
    "            k += elem_dof_p\n",
    "        end\n",
    "    elseif p_is_tensor && !p_is_continuous && Pp > 1\n",
    "        elem_dof_p = Pp^2\n",
    "        idx_p = zeros(Int64, elem_dof_p, num_elem) # Solution connectivity for p\n",
    "        k = 0\n",
    "        for i=1:num_elem\n",
    "            for j=1:elem_dof_p\n",
    "                idx_p[j, i] = j + k\n",
    "            end\n",
    "            k += elem_dof_p\n",
    "        end\n",
    "    elseif p_is_tensor && p_is_continuous && Pp > 1\n",
    "        nodes_p = zeros(Int64, 2)\n",
    "        for d=1:2\n",
    "            nodes_p[d] = n_elems[d]*(Pp-1) + 1\n",
    "        end\n",
    "        idx_p = zeros(Int64, Pp * Pp, num_elem) # Solution connectivity for p\n",
    "        for i=1:n_elems[1]\n",
    "            for j=1:n_elems[2]\n",
    "                ele = (j-1)*n_elems[1] + i\n",
    "                for ii=1:Pp\n",
    "                    for jj=1:Pp\n",
    "                        idx_p[(ii-1) * Pp + jj, ele] = ((j-1) * (Pp - 1) + ii-1) * nodes_p[1] + (i-1) * (Pp - 1) + jj\n",
    "                    end\n",
    "                end\n",
    "            end\n",
    "        end\n",
    "    else\n",
    "        error(\"Unsupported dof for pressure field\");\n",
    "    end\n",
    "    \n",
    "    return idx_p\n",
    "end\n",
    "\n",
    "struct FEIndices\n",
    "    P::Int\n",
    "    Pp::Int\n",
    "    nx::Int\n",
    "    ny::Int\n",
    "    idx_x::Matrix\n",
    "    idx_u::Matrix\n",
    "    idx_p::Matrix\n",
    "    idx_b::Vector\n",
    "    idx_r::Vector\n",
    "    idx_t::Vector\n",
    "    idx_l::Vector\n",
    "    nodes_u::Vector\n",
    "    Er_x::Vector\n",
    "    Er_u::Vector\n",
    "    Er_p::Vector\n",
    "    p_is_continuous::Bool\n",
    "    p_is_tensor::Bool\n",
    "    function FEIndices(P, Pp, num_comp, nx, ny, p_is_continuous, p_is_tensor)\n",
    "        idx_x, idx_u, idx_b, idx_r, idx_t, idx_l, nodes_u = GetConnectivity(P, nx, ny)\n",
    "        idx_p = GetConnectivityPressure(Pp, nx, ny, p_is_continuous, p_is_tensor)\n",
    "        # number of nodes\n",
    "        dof_u = maximum(idx_u)\n",
    "        dof_p = maximum(idx_p)\n",
    "        dof_x = maximum(idx_x)\n",
    "        # element dof\n",
    "        elem_dof_u = size(idx_u, 1)\n",
    "        elem_dof_p = size(idx_p, 1)\n",
    "        elem_dof_x = size(idx_x, 1)\n",
    "        num_elem = size(idx_x, 2)\n",
    "        # element restriction operator\n",
    "        Er_x = []\n",
    "        Er_u = []\n",
    "        Er_p = []\n",
    "        idx_p = idx_p .+ num_comp * dof_u\n",
    "        for e = 1:num_elem\n",
    "            Lx = spzeros(Int64, elem_dof_x, dof_x)\n",
    "            Lu = spzeros(Int64, elem_dof_u, dof_u)\n",
    "            L = spzeros(Int64, elem_dof_u * num_comp, dof_p)\n",
    "            Lp = zeros(Int64, elem_dof_p, num_comp*dof_u + dof_p)\n",
    "            for i=1:elem_dof_u\n",
    "                if i < elem_dof_x + 1\n",
    "                    Lx[i, idx_x[i, e] ] = 1\n",
    "                end\n",
    "                Lu[i, idx_u[i, e] ] = 1\n",
    "            end\n",
    "            \n",
    "            for i=1:elem_dof_p\n",
    "                Lp[i, idx_p[i, e] ] = 1\n",
    "            end\n",
    "            \n",
    "            Lx = kron(I(2), Lx)\n",
    "            Lu = kron(I(num_comp), Lu)\n",
    "            Lu = hcat(Lu, L)\n",
    "            push!(Er_x, Lx)\n",
    "            push!(Er_u, Lu)\n",
    "            push!(Er_p, Lp)\n",
    "        end\n",
    "        \n",
    "        new(P, Pp, nx, ny, idx_x, idx_u, idx_p, idx_b, idx_r, idx_t, idx_l, nodes_u, Er_x, Er_u, Er_p, p_is_continuous, p_is_tensor)\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "8ce01307",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GetCoordElem (generic function with 1 method)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function GetCoordMesh(mesh, nelx, nely, aspect_ratio)\n",
    "    \"\"\" This function returns the physical coordinates of the nodes.\n",
    "    Input:\n",
    "    ------\n",
    "    nelx:   integer\n",
    "            number of elements in the x direction.\n",
    "    nely:   integer\n",
    "            number of elements in the y direction.\n",
    "    mesh: can be unifrom or nonuniform\n",
    "\n",
    "    Output:\n",
    "    -------\n",
    "    x:      float (1d array)\n",
    "            the coordinate of the node in the x direction\n",
    "    y:      float (1d array)\n",
    "            the coordinate of the node in the y direction\n",
    "    The geometry we are working on is like the following.\n",
    "    (for nelx = 2, nely = 2)\n",
    "    6---------7----------8\n",
    "    |         |   (3)    |\n",
    "    |   (2)   |      ----5\n",
    "    |      ---4-----/    |\n",
    "    3-----/   |   (1)    |\n",
    "    |         |      ----2\n",
    "    |   (0)   |     /\n",
    "    |     ----1----/\n",
    "    0----/\n",
    "    There are 4 elements (numbering in parenthesis), and 9 nodes.\n",
    "    This function returns x,y as 9x2 array for the above mesh.\n",
    "    \"\"\"\n",
    "    nodex = nelx + 1\n",
    "    nodey = nely + 1\n",
    "    numnodes = nodex*nodey\n",
    "    # interior nodes for random mesh\n",
    "    interiornodex = nelx - 1\n",
    "    interiornodey = nely - 1\n",
    "    interiornodes = interiornodex*interiornodey\n",
    "\n",
    "    hx = 1/nelx\n",
    "    hy = 1/(nely*aspect_ratio)\n",
    "    h = minimum([hx hy])\n",
    "    \n",
    "    if (aspect_ratio <1)\n",
    "        error!(\"aspect_ratio must be bigger than 1\")\n",
    "    end\n",
    "\n",
    "    # Divide [0,1] by nodex (mesh in the x direction)\n",
    "    x0 = LinRange(0, 1, nodex)\n",
    "    if mesh == \"uniform\"\n",
    "        y0 = 0.0*x0 # the bottom geometry line  \n",
    "        y = zeros(numnodes, 1)\n",
    "        for i=1:nodex\n",
    "            # Divide [0,1] by nodey (mesh in the y direction)\n",
    "            y1 = LinRange(y0[i], 1/aspect_ratio, nodey)\n",
    "            for j=1:nodey\n",
    "                y[i + (j-1)*nodex] = y1[j]   # collection of y\n",
    "            end\n",
    "        end\n",
    "        x = zeros(numnodes, 1)\n",
    "        for i=1:nodey\n",
    "            for j=1:nodex\n",
    "                x[j + (i-1)*nodex] = x0[j]   # collection of x\n",
    "            end\n",
    "        end\n",
    "\n",
    "    elseif mesh == \"random\"\n",
    "        y0 = 0.0*x0 # the bottom geometry line  \n",
    "        y = zeros(numnodes, 1)\n",
    "        for i=1:nodex\n",
    "            # Divide [0,1] by nodey (mesh in the y direction)\n",
    "            y1 = LinRange(y0[i], 1/aspect_ratio, nodey)\n",
    "            for j=1:nodey\n",
    "                y[i + (j-1)*nodex] = y1[j]   # collection of y\n",
    "            end\n",
    "        end\n",
    "        x = zeros(numnodes, 1)\n",
    "        for i=1:nodey\n",
    "            for j=1:nodex\n",
    "                x[j + (i-1)*nodex] = x0[j]   # collection of x\n",
    "            end\n",
    "        end\n",
    "        Random.seed!(1234)\n",
    "        randnodes_x = rand(interiornodes,1)*hx/2 .- hx/4\n",
    "        randnodes_y = rand(interiornodes,1)*hy/2 .- hy/4\n",
    "        # perturb the (x,y) of interior nodes\n",
    "        for i=1:interiornodey\n",
    "            for j=1:interiornodex\n",
    "                x[i*(nodex) + j+1] = x[i*(nodex) + j+1] - randnodes_x[j+(i-1)*interiornodex]\n",
    "                y[i*(nodex) + j+1] = y[i*(nodex) + j+1] - randnodes_y[j+(i-1)*interiornodex]\n",
    "            end\n",
    "        end\n",
    "        \n",
    "    elseif mesh == \"trapezoid\"\n",
    "        y0 = 0.0*x0 # the bottom geometry line  \n",
    "        y = zeros(numnodes, 1)\n",
    "        for i=1:nodex\n",
    "            # Divide [0,1] by nodey (mesh in the y direction)\n",
    "            y1 = LinRange(y0[i], 1, nodey)\n",
    "            for j=1:nodey\n",
    "                y[i + (j-1)*nodex] = y1[j]   # collection of y\n",
    "            end\n",
    "        end\n",
    "        x = zeros(numnodes, 1)\n",
    "        for i=1:nodey\n",
    "            for j=1:nodex\n",
    "                x[j + (i-1)*nodex] = x0[j]   # collection of x\n",
    "            end\n",
    "        end\n",
    "        # perturb the (x,y) of interior nodes\n",
    "        for i=1:interiornodey\n",
    "            for j=1:interiornodex\n",
    "                #x[i*(nodex) + j+1] = x[i*(nodex) + j+1] - randnodes[j+(i-1)*interiornodex]\n",
    "                y[i*(nodex) + j+1] = y[i*(nodex) + j+1] - (-1)^j *(h/4)\n",
    "            end\n",
    "        end\n",
    "\n",
    "    else\n",
    "        error(\"Enter one of the mesh option: 'unifrom', 'trapezoid', 'random' \")\n",
    "    end\n",
    "    \n",
    "    #\n",
    "    return x, y\n",
    "\n",
    "end\n",
    "\n",
    "function plotmesh(mesh, xc, yc, idx_x)\n",
    "\n",
    "    p = plot()\n",
    "    nodes = [1 2 4 3 1]\n",
    "    xx = zeros(5,1)\n",
    "    yy = zeros(5,1)\n",
    "    num_elem = size(idx_x, 2)\n",
    "    for i = 1:num_elem\n",
    "        for j =1:5\n",
    "        xx[j] = xc[idx_x[nodes[j],i]] \n",
    "        yy[j] = yc[idx_x[nodes[j],i]] \n",
    "        end\n",
    "        plot!(p,xx,yy,color=\"black\",legend=false)\n",
    "    end\n",
    "    \n",
    "    return p\n",
    "end\n",
    "\n",
    "function GetCoordElem(xc, yc, Er_x, e)\n",
    "    \"\"\"\n",
    "    This functions returns coordinate of element \"E\"\n",
    "    Input:\n",
    "    x,y,z: coordinate of all nodes of a mesh\n",
    "    IENn: connectivity of nodes\n",
    "    \n",
    "    Return:\n",
    "    coord_E: coordinate of element e as a (8,1) array\n",
    "    which is [x;y] coordinates of the 4 nodes\n",
    "    \"\"\"\n",
    "    \n",
    "    return Er_x[e]*[xc; yc]\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "679cc47e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GetL2ErrorDisc (generic function with 1 method)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function GetQdata(Coord_E, Bx::FEBasis)\n",
    "    J1 = Bx.D*Coord_E\n",
    "    J2 = reshape(J1, Bx.Q^2, :)\n",
    "    J = zeros(Bx.Q^2, 2, 2)\n",
    "    detJ = zeros(Bx.Q^2)\n",
    "    dXdx_T = zeros(Bx.Q^2, 2, 2)\n",
    "    for i=1:Bx.Q^2\n",
    "        J[i,:,:] = reshape(J2[i,:], 2, 2)\n",
    "        detJ[i] = det(J[i,:,:])\n",
    "        dXdx_T[i,:,:] = I/J[i,:,:]\n",
    "    end\n",
    "    \n",
    "    # quadrature pts in elmenet E\n",
    "    q = Bx.B * Coord_E\n",
    "    \n",
    "    wdetJ = Bx.w_ref .* detJ;\n",
    "    \n",
    "    return q, wdetJ, dXdx_T\n",
    "end\n",
    "\n",
    "function GetRhsLocal(kappa, mu, wdetJ, q, f, Bu::FEBasis)\n",
    "    \n",
    "    Wu = diagm([wdetJ; wdetJ])\n",
    "    xq = reshape(q, Bu.Q^2,:)\n",
    "    Feu = Bu.B' * Wu * f(kappa, mu, xq[:,1], xq[:,2])  # v^T * f\n",
    "    Fep = zeros(size(Bu.Bp,2))\n",
    "    return Feu, Fep\n",
    "end\n",
    "\n",
    "function GetLocalOperator(kappa, mu, wdetJ, dXdx_T, Bu::FEBasis)\n",
    "    \n",
    "    Wp = diagm(wdetJ)\n",
    "    Ws = diagm([wdetJ; wdetJ; wdetJ; wdetJ]);\n",
    "    \n",
    "    # sigma = -p I + 2 mu ed; where, ed = e - 1/3 * trace(e) * I, I know it should be 1/2 beacuse is 2D, anyway ...\n",
    "    # e = 0.5*(grad_u + grad_u^T)\n",
    "    grad_v = zeros(2 * Bu.num_comp * Bu.Q^2, size(Bu.D,2))\n",
    "    div_v = zeros(2 * Bu.num_comp * Bu.Q^2, size(Bu.Bp,2))\n",
    "    q_div = zeros(Bu.Q^2, size(Bu.D,2))\n",
    "    qp  = zeros(Bu.Q^2, size(Bu.Bp,2))\n",
    "    for i=1:Bu.Q^2\n",
    "        D1 = zeros(2, size(Bu.D,2))\n",
    "        D1[1,:] = Bu.D[i,:]\n",
    "        D1[2,:] = Bu.D[i + Bu.Q^2,:]\n",
    "        grad_u1 = dXdx_T[i,:,:] * D1\n",
    "        D2 = zeros(2, size(Bu.D,2))\n",
    "        D2[1,:] = Bu.D[i + 2*Bu.Q^2,:]\n",
    "        D2[2,:] = Bu.D[i + 3*Bu.Q^2,:]\n",
    "        grad_u2 = dXdx_T[i,:,:] * D2\n",
    "        # tr(e) = div(u)\n",
    "        tr_e = grad_u1[1,:] + grad_u2[2,:]\n",
    "        # ed = e - 1/3 * trace(e) * I\n",
    "        ed11 = grad_u1[1,:] - 1/3. * tr_e\n",
    "        ed12 = (grad_u1[2,:] + grad_u2[1,:]) / 2.\n",
    "        ed22 = grad_u2[2,:] - 1/3. * tr_e\n",
    "        Ed1 = dXdx_T[i,:,:]' * vcat(ed11', ed12')\n",
    "        Ed2 = dXdx_T[i,:,:]' * vcat(ed12', ed22')\n",
    "\n",
    "        p1 = Bu.Bp[i,:]\n",
    "        p12 = 0. * Bu.Bp[i,:]\n",
    "        P1 = dXdx_T[i,:,:]' * vcat(p1', p12')\n",
    "        P2 = dXdx_T[i,:,:]' * vcat(p12', p1')\n",
    "        # 2 mu (grad(v), e)\n",
    "        grad_v[i,:] = 2 * mu * Ed1[1,:]\n",
    "        grad_v[i + Bu.Q^2,:] = 2 * mu * Ed1[2,:]\n",
    "        grad_v[i + 2*Bu.Q^2,:] = 2 * mu * Ed2[1,:]\n",
    "        grad_v[i + 3*Bu.Q^2,:] = 2 * mu * Ed2[2,:]\n",
    "        # -(div(v), p)\n",
    "        div_v[i,:] = -P1[1,:]\n",
    "        div_v[i + Bu.Q^2,:] = -P1[2,:]\n",
    "        div_v[i + 2*Bu.Q^2,:] = -P2[1,:]\n",
    "        div_v[i + 3*Bu.Q^2,:] = -P2[2,:]\n",
    "        # -(q, div(u))\n",
    "        q_div[i, :] = -tr_e\n",
    "        # -(q, p)/kappa\n",
    "        qp[i,:] = -Bu.Bp[i,:] / kappa\n",
    "        \n",
    "    end\n",
    "    \n",
    "    Ke11 = Bu.D' * Ws * grad_v\n",
    "    Ke12 = Bu.D' * Ws * div_v\n",
    "    Ke21 = Bu.Bp' * Wp * q_div\n",
    "    Ke22 = Bu.Bp' * Wp * qp\n",
    "    \n",
    "    return Ke11, Ke12, Ke21, Ke22\n",
    "end\n",
    "\n",
    "function GetDirichletBCsIndex(num_comp, Ind::FEIndices)\n",
    "    num_nodes = Ind.nodes_u[1] * Ind.nodes_u[2]\n",
    "    bc_idx = Int64[]\n",
    "    for c=0:num_comp -1\n",
    "        # bottom\n",
    "        for i=1:Ind.nodes_u[1]\n",
    "            append!(bc_idx, Ind.idx_b[i] .+ c * num_nodes)\n",
    "        end\n",
    "        # right\n",
    "        for j=2:Ind.nodes_u[2]-1\n",
    "            append!(bc_idx, Ind.idx_r[j] .+ c * num_nodes)\n",
    "        end\n",
    "        # top\n",
    "        for i=1:Ind.nodes_u[1]\n",
    "            append!(bc_idx, Ind.idx_t[i] .+ c * num_nodes)\n",
    "        end\n",
    "        # left\n",
    "        for j=2:Ind.nodes_u[2]-1\n",
    "            append!(bc_idx, Ind.idx_l[j] .+ c * num_nodes)\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    return bc_idx\n",
    "end\n",
    "\n",
    "function GetNodalCoordinate(xc, yc, Q, Ind::FEIndices)\n",
    "    \"\"\"\n",
    "    return x for plotting the results in 2D\n",
    "    \"\"\"\n",
    "    num_elem = size(Ind.idx_u,2)\n",
    "    global_dof = maximum(Ind.idx_p);\n",
    "    global_dof_u = 2 * maximum(Ind.idx_u);\n",
    "    global_dof_p = global_dof - global_dof_u;\n",
    "    \n",
    "    _, _, _, BL, _ = febasis2D(2, Q, 2, \"GLL\");\n",
    "    xu = spzeros(global_dof, 1)\n",
    "    mult_u = spzeros(global_dof, 1)\n",
    "    for e=1:num_elem\n",
    "        # Get coordinate of each element\n",
    "        Coord_E = GetCoordElem(xc, yc, Ind.Er_x, e);\n",
    "        xe = BL * Coord_E\n",
    "        xu = xu + Ind.Er_u[e]' * xe\n",
    "        mult_u += Ind.Er_u[e]' * Ind.Er_u[e] * ones(global_dof)\n",
    "    end\n",
    "    xu = xu ./ mult_u\n",
    "    xu[1:global_dof_u]\n",
    "end\n",
    "\n",
    "function GetFEMixed(kappa, mu, xc, yc, Ind::FEIndices, Bx::FEBasis, Bu::FEBasis, f, bc_idx, u_bc)\n",
    "\n",
    "    num_elem = size(Ind.idx_u,2)\n",
    "    global_dof = maximum(Ind.idx_p)\n",
    "    global_dof_u = 2 * maximum(Ind.idx_u)\n",
    "    F = spzeros(global_dof, 1)\n",
    "    K = spzeros(global_dof, global_dof)\n",
    "    \n",
    "    for e=1:num_elem\n",
    "        # Get coordinate of each element\n",
    "        Coord_E = GetCoordElem(xc, yc, Ind.Er_x, e);\n",
    "        # Update quadrature\n",
    "        q, wdetJ, dXdx = GetQdata(Coord_E, Bx);\n",
    "        \n",
    "        Ke11, Ke12, Ke21, Ke22 = GetLocalOperator(kappa, mu, wdetJ, dXdx, Bu)\n",
    "        Feu, Fep = GetRhsLocal(kappa, mu, wdetJ, q, f, Bu);  # v^T * f\n",
    "\n",
    "        # Assemble\n",
    "        K = K + Ind.Er_u[e]' * Ke11 * Ind.Er_u[e] + Ind.Er_u[e]' * Ke12 * Ind.Er_p[e] +\n",
    "                Ind.Er_p[e]' * Ke21 * Ind.Er_u[e] + Ind.Er_p[e]' * Ke22 * Ind.Er_p[e];\n",
    "        F = F + Ind.Er_u[e]' * Feu + Ind.Er_p[e]' * Fep\n",
    "\n",
    "    end\n",
    "    \n",
    "    F[bc_idx].= u_bc\n",
    "    K[bc_idx,:] .= 0.; #K[:, bc_idx] .= 0.\n",
    "    for i=1:length(bc_idx)\n",
    "        K[bc_idx[i], bc_idx[i]] = 1.\n",
    "    end\n",
    "    # U=[u, p]\n",
    "    U = K \\ F\n",
    "    \n",
    "   return U\n",
    "end\n",
    "\n",
    "function GetL2Error(kappa, xc, yc, Ind::FEIndices, Bx::FEBasis, Bu::FEBasis, Uh, uex, pex)\n",
    "    \"\"\"This function returns L2 error \n",
    "    (sum_{e=1:num_elem} (int_e (uh - uex)^2 dx) )^0.5 \n",
    "    \"\"\"\n",
    "\n",
    "    num_elem = size(Ind.idx_u,2)\n",
    "\n",
    "    e_u = zeros(num_elem)\n",
    "    e_p = zeros(num_elem)\n",
    "    for e=1:num_elem\n",
    "        # Get coordinate of each element\n",
    "        Coord_E = GetCoordElem(xc, yc, Ind.Er_x, e);\n",
    "        # Update quadrature\n",
    "        q, wdetJ, _ = GetQdata(Coord_E, Bx);\n",
    "        \n",
    "        wdetJu = [wdetJ; wdetJ]\n",
    "        \n",
    "        # dh and ph are the nodal values for u and p for element e\n",
    "        dh = Ind.Er_u[e] * Uh\n",
    "        ph = Ind.Er_p[e] * Uh\n",
    "        # Compute uh(x) and ph(x) in quadrature x=q in L2 space \n",
    "        u_num = Bu.B * dh\n",
    "        p_num = Bu.Bp * ph\n",
    "        \n",
    "        xq = reshape(q, Q^2,:)\n",
    "        u_ex = uex(xq[:,1], xq[:,2])\n",
    "        p_ex = pex(kappa, xq[:,1], xq[:,2])\n",
    "\n",
    "        err_u = vec((u_num - u_ex))\n",
    "        err_p = vec((p_num - p_ex))\n",
    "        # \\int (u_num - u_ex)^2 dx ==> on element e\n",
    "        e_u[e] = wdetJu' * (err_u .* err_u)\n",
    "        e_p[e] = wdetJ' * (err_p .* err_p)\n",
    "\n",
    "    end\n",
    "\n",
    "    return sqrt(sum(e_u)), sqrt(sum(e_p))\n",
    "end\n",
    "    \n",
    "function GetL2ErrorDisc(kappa, xc, yc, Ind::FEIndices, Bx::FEBasis, Bu::FEBasis, Uh, uex, pex)\n",
    "    \"\"\"This function returns L2 error \n",
    "    (sum_{e=1:num_elem} (int_e (uh - uex)^2 dx) )^0.5 \n",
    "    \"\"\"\n",
    "\n",
    "    num_elem = size(Ind.idx_u,2)\n",
    "    global_dof = maximum(Ind.idx_p)\n",
    "    global_dof_u = Bu.num_comp * maximum(Ind.idx_u)\n",
    "        \n",
    "    e_U = zeros(global_dof)\n",
    "        \n",
    "    for e=1:num_elem\n",
    "        # Get coordinate of each element\n",
    "        Coord_E = GetCoordElem(xc, yc, Ind.Er_x, e);\n",
    "        # Update quadrature\n",
    "        q, wdetJ = GetQdata(Coord_E, Bx);\n",
    "\n",
    "        # dh and ph are the nodal values for u and p for element e\n",
    "        dh = Ind.Er_u[e] * Uh\n",
    "        ph = Ind.Er_p[e] * Uh\n",
    "        # Compute uh(x) and ph(x) in quadrature x=q in L2 space \n",
    "        u_num = Bu.B * dh\n",
    "        p_num = Bu.Bp * ph\n",
    "        \n",
    "        xq = reshape(q, Q^2,:)\n",
    "        u_ex = uex(xq[:,1], xq[:,2])\n",
    "        p_ex = pex(kappa, xq[:,1], xq[:,2])\n",
    "\n",
    "        err_u = vec((u_num - u_ex))\n",
    "        err_p = vec((p_num - p_ex))\n",
    "            \n",
    "        Wu = diagm([wdetJ; wdetJ])\n",
    "        Wp = diagm(wdetJ)\n",
    "        er1 = Bu.B' * Wu * (err_u .* err_u)\n",
    "        er2 = Bu.Bp' * Wp * (err_p .* err_p)\n",
    "        \n",
    "        e_U = e_U + Ind.Er_u[e]' * er1 + Ind.Er_p[e]' * er2\n",
    "\n",
    "    end\n",
    "        \n",
    "\n",
    "    return sqrt(sum(e_U[1:global_dof_u])), sqrt(sum(e_U[global_dof_u+1:end]))\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e46e1039",
   "metadata": {},
   "source": [
    "# Inf-Sup test\n",
    "To get the inf-sup cosntant we need to solve (see eq (3.2) of Arnold and Marie 2009)\n",
    "\n",
    "$$\n",
    "<u,v>_V + b(v,p) + b(u,q) = -\\lambda <p,q>_Q \\quad \\text{for all} \\quad v,q \\in V\\times Q\n",
    "$$\n",
    "Then, the inf-sup constant, $\\beta$ is\n",
    "$$\n",
    "\\beta = \\sqrt{\\lambda_{min}}\n",
    "$$\n",
    "\n",
    "We can create\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix} \\boldsymbol H &\\boldsymbol B^T\\\\\\boldsymbol B& \\boldsymbol 0 \\end{bmatrix} \\begin{bmatrix}\\boldsymbol U\\\\ \\boldsymbol P \\end{bmatrix}=  -\\lambda \\begin{bmatrix} \\boldsymbol 0 & \\boldsymbol 0\\\\ \\boldsymbol 0 &\\boldsymbol C \\end{bmatrix}  \\begin{bmatrix}\\boldsymbol U\\\\ \\boldsymbol P \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "where $\\boldsymbol H = <\\boldsymbol u,\\boldsymbol v>,\\boldsymbol B = b(\\boldsymbol u,q), \\boldsymbol B^T = b(\\boldsymbol v,p), \\boldsymbol C = <p,q>$.\n",
    "\n",
    "By solving above equation\n",
    "$$\n",
    "\\boldsymbol H \\boldsymbol U = -\\boldsymbol B^T \\boldsymbol P , \\\\\n",
    "\\boldsymbol B \\boldsymbol U = -\\lambda \\boldsymbol C \\boldsymbol P\n",
    "$$\n",
    "from the first equation we get $\\boldsymbol U = -\\boldsymbol H^{-1} \\boldsymbol B^T \\boldsymbol P$ and replace it in second equation gives\n",
    "$$\n",
    "(\\boldsymbol B \\boldsymbol H^{-1} \\boldsymbol{B}^T ) \\boldsymbol P = \\lambda \\boldsymbol C \\boldsymbol P\n",
    "$$\n",
    "\n",
    "$\\sqrt{\\lambda_{min}}$ in above equation is infsup constant.\n",
    "$$\n",
    "\\boldsymbol H =<\\boldsymbol v,\\boldsymbol u>_{H1}= \\int \\boldsymbol v \\boldsymbol u + \\int \\nabla\\boldsymbol v \\nabla \\boldsymbol u,\\\\\n",
    "\\boldsymbol C = <p,q>_{L^2} = \\int pq\n",
    "$$\n",
    "\n",
    "To find coercivity constant we solve\n",
    "\n",
    "$$\n",
    "a(u,v) + b(v,p) + b(u,q) = \\lambda <u,v>_V \\quad \\text{for all} \\quad v,q \\in V\\times Q\n",
    "$$\n",
    "Then, the coercivity constant, $\\alpha$ is\n",
    "$$\n",
    "\\alpha = \\min{|\\lambda_{min}|}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "6b29b747",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GetInfSup (generic function with 1 method)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function GetInfSupLocalOperator(q, wdetJ, dXdx_T, Bu::FEBasis, P1_config)\n",
    "    \n",
    "    Wp = diagm(wdetJ)\n",
    "    Wm = diagm([wdetJ; wdetJ]);\n",
    "    Ws = diagm([wdetJ; wdetJ; wdetJ; wdetJ]);\n",
    "    \n",
    "    if P1_config == \"physical\"\n",
    "        basis_p = P1_basis_phy(q, Bu.Q)\n",
    "    else\n",
    "        basis_p = Bu.Bp\n",
    "    end\n",
    "    grad_v = zeros(2 * Bu.num_comp * Bu.Q^2, size(Bu.D,2))\n",
    "    div_v = zeros(2 * Bu.num_comp * Bu.Q^2, size(Bu.Bp,2))\n",
    "    q_div = zeros(Bu.Q^2, size(Bu.D,2))\n",
    "    qp  = zeros(Bu.Q^2, size(basis_p, 2))\n",
    "    for i=1:Bu.Q^2\n",
    "        D1 = zeros(2, size(Bu.D,2));\n",
    "        D1[1,:] = Bu.D[i,:]\n",
    "        D1[2,:] = Bu.D[i + Bu.Q^2,:]\n",
    "        grad_u1 = dXdx_T[i,:,:] * D1\n",
    "        D2 = zeros(2, size(Bu.D,2));\n",
    "        D2[1,:] = Bu.D[i + 2*Bu.Q^2,:]\n",
    "        D2[2,:] = Bu.D[i + 3*Bu.Q^2,:]\n",
    "        grad_u2 = dXdx_T[i,:,:] * D2\n",
    "        # tr(e) = div(u)\n",
    "        tr_e = grad_u1[1,:] + grad_u2[2,:]\n",
    "        G1 = dXdx_T[i,:,:]' * grad_u1\n",
    "        G2 = dXdx_T[i,:,:]' * grad_u2\n",
    "\n",
    "        # (grad(v), grad(u))\n",
    "        grad_v[i,:] = G1[1,:]\n",
    "        grad_v[i + Bu.Q^2,:] = G1[2,:]\n",
    "        grad_v[i + 2*Bu.Q^2,:] = G2[1,:]\n",
    "        grad_v[i + 3*Bu.Q^2,:] = G2[2,:]\n",
    "        # B = (q, div(u))\n",
    "        q_div[i, :] = tr_e\n",
    "        # C = (q, p)\n",
    "        qp[i,:] = basis_p[i,:]\n",
    "        \n",
    "    end\n",
    "    # <v,u>_H1\n",
    "    He = Bu.B' * Wm * Bu.B + Bu.D' * Ws * grad_v\n",
    "    # b(u, q)\n",
    "    Be = basis_p' * Wp * q_div\n",
    "    # <p,q>_L2\n",
    "    Ce = basis_p' * Wp * qp\n",
    "    \n",
    "    return He, Be, Ce\n",
    "end\n",
    "\n",
    "\n",
    "function GetInfSup(xc, yc, Ind::FEIndices, Bx::FEBasis, Bu::FEBasis, P1_config)\n",
    "\n",
    "    num_elem = size(Ind.idx_u,2)\n",
    "    global_dof = maximum(Ind.idx_p)\n",
    "    global_dof_u = 2 * maximum(Ind.idx_u)\n",
    "    \n",
    "    K1 = spzeros(global_dof, global_dof)\n",
    "    \n",
    "    for e=1:num_elem\n",
    "        # Get coordinate of each element\n",
    "        Coord_E = GetCoordElem(xc, yc, Ind.Er_x, e);\n",
    "        # Update quadrature\n",
    "        q, wdetJ, dXdx_T = GetQdata(Coord_E, Bx);\n",
    "        \n",
    "        He, Be, Ce = GetInfSupLocalOperator(q, wdetJ, dXdx_T, Bu, P1_config)\n",
    "\n",
    "        # Assemble\n",
    "        K1 = K1 + Ind.Er_u[e]' * He * Ind.Er_u[e] +\n",
    "                Ind.Er_p[e]' * Be * Ind.Er_u[e] + Ind.Er_p[e]' * Ce * Ind.Er_p[e];\n",
    "\n",
    "    end\n",
    "    \n",
    "    H = K1[1:global_dof_u,1:global_dof_u];\n",
    "    C = K1[global_dof_u + 1:end, global_dof_u + 1:end]\n",
    "    B = K1[global_dof_u + 1:end, 1:global_dof_u]\n",
    "    \n",
    "    LHS1 = B * (H \\ I) * B'\n",
    "    RHS1 = C\n",
    "    D = eigvals(LHS1, RHS1)\n",
    "    D = real(D)\n",
    "    lambda1_min = minimum(D) \n",
    "    # inf-sup constant\n",
    "    beta = sqrt(lambda1_min)\n",
    "     \n",
    "   return beta\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "14c3210d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(L2_error_u, L2_error_p, beta) = (9.466582004640185e-5, 0.004167826125253, 0.9626665946369111)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(9.466582004640185e-5, 0.004167826125253, 0.9626665946369111)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# P is number of nodes in 1D, P=2 means Q1 element\n",
    "# Pp is number of nodes in 1D if the pressure is tensor basis, otherwise is number of nodes in 2D\n",
    "P, Pp, Qmode, p_is_continuous, p_is_tensor = 3, 2, \"GAUSS\", true, true\n",
    "Q, num_comp = P, 2\n",
    "Bx = FEBasis(2, Pp, Q, 2, Qmode, p_is_tensor);\n",
    "Bu = FEBasis(P, Pp, Q, num_comp, Qmode, p_is_tensor);\n",
    "nx, ny, mesh, aspect_ratio= 4, 4, \"uniform\", 1\n",
    "xc, yc = GetCoordMesh(mesh, nx, ny, aspect_ratio);\n",
    "Ind = FEIndices(P, Pp, num_comp, nx, ny, p_is_continuous, p_is_tensor);\n",
    "lambda, mu = 3, 1;\n",
    "kappa = lambda + 2*mu/3;\n",
    "\n",
    "function uex(x, y)\n",
    "    A0 = 1e-3\n",
    "    s = pi*2\n",
    "    x0 = pi/2\n",
    "    y0 = x0\n",
    "    ux = @. A0 * sin(s*x - x0) * sin(s*y - y0)\n",
    "    uy = @. 2*ux \n",
    "    \n",
    "    return [ux; uy]\n",
    "end\n",
    "\n",
    "function pex(kappa, x, y)\n",
    "    A0 = 1e-3\n",
    "    s = pi*2\n",
    "    x0 = pi/2\n",
    "    y0 = x0\n",
    "    ux_x = @. A0 * s * cos(s*x - x0) * sin(s*y - y0)\n",
    "    ux_y = @. A0 * s * sin(s*x - x0) * cos(s*y - y0)\n",
    "    uy_y = @. 2 * ux_y\n",
    "    \n",
    "    p = @. -kappa * (ux_x + uy_y)\n",
    "    return p\n",
    "end\n",
    "\n",
    "function f(kappa, mu, x, y)\n",
    "    A0 = 1e-3\n",
    "    s = pi*2\n",
    "    x0 = pi/2\n",
    "    y0 = x0\n",
    "    ux = @. A0 * sin(s*x - x0) * sin(s*y - y0)\n",
    "    uy = @. 2*ux \n",
    "    \n",
    "    ux_xx = @. -s^2 * ux\n",
    "    ux_yy = @. -s^2 * ux\n",
    "    uy_xx = @. 2 * ux_xx\n",
    "    uy_yy = @. 2 * ux_yy\n",
    "    ux_xy = @. A0 * s^2 * cos(s*x - x0) * cos(s*y - y0)\n",
    "    uy_yx = @. 2 * ux_xy\n",
    "    \n",
    "    fx = @. -mu * (ux_xx + ux_yy) - (mu/3 + kappa) * (ux_xx + uy_yx)\n",
    "    fy = @. -mu * (uy_xx + uy_yy) - (mu/3 + kappa) * (ux_xy + uy_yy) \n",
    "    return [fx; fy]\n",
    "end\n",
    "\n",
    "\n",
    "# Dirichlet BCs\n",
    "bc_idx = GetDirichletBCsIndex(Bu.num_comp, Ind);\n",
    "x = GetNodalCoordinate(xc, yc, Bu.Q, Ind);\n",
    "x_bc = x[bc_idx]\n",
    "x_bc = reshape(x_bc,:,2)\n",
    "u_bc = uex(x_bc[:,1], x_bc[:,2])\n",
    "\n",
    "uh = GetFEMixed(kappa, mu, xc, yc, Ind, Bx, Bu, f, bc_idx, u_bc);\n",
    "L2_error_u, L2_error_p = GetL2Error(kappa, xc, yc, Ind, Bx, Bu, uh, uex, pex);\n",
    "P1_config = \"\"\n",
    "beta = GetInfSup(xc, yc, Ind, Bx, Bu, P1_config);\n",
    "@show L2_error_u, L2_error_p, beta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "b549c33a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(L2_error_u, L2_error_p) = (9.466582004640185e-5, 0.004167826125253)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(9.466582004640185e-5, 0.004167826125253)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L2_error_u, L2_error_p = GetL2ErrorDisc(kappa, xc, yc, Ind, Bx, Bu, uh, uex, pex);\n",
    "@show L2_error_u, L2_error_p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "e8988c78",
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xx = reshape(x, :,2)\n",
    "err = uh[1:length(x)] - uex(xx[:,1], xx[:,2])\n",
    "ee = reshape(err, :, 2)\n",
    "ex = reshape(ee[:,1], Ind.nodes_u[1], Ind.nodes_u[2]) \n",
    "ey = reshape(ee[:,2], Ind.nodes_u[1], Ind.nodes_u[2]) \n",
    "x1 = reshape(xx[:,1], Ind.nodes_u[1], Ind.nodes_u[2]) \n",
    "x2 = reshape(xx[:,2], Ind.nodes_u[1], Ind.nodes_u[2]) \n",
    "p1=contourf(x1[:,1], x2[1,:], ex', color=:turbo, levels=10, title=\"Galerkin error, componenet x: $(norm(ee[:,1]))\");\n",
    "p2=contourf(x1[:,1], x2[1,:], ey', color=:turbo, levels=10, title=\"Galerkin error, componenet y: $(norm(ee[:,2]))\");\n",
    "scatter!(xx[:,1], xx[:,2], marker=:star, label = \"nodes\")\n",
    "plot(p1,p2, layout=(2,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c2313e15",
   "metadata": {},
   "source": [
    "# <font color='red'> Matrix-Free approach"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "ab0e0152",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Compute_df1dg0 (generic function with 1 method)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# This is like libCEED/Ratel Residual Qfunction\n",
    "function Compute_f0(kappa, mu, Q, num_comp, wdetJ, q, uq, pq, f)\n",
    "    \n",
    "    Wu = diagm([wdetJ; wdetJ])\n",
    "    xq = reshape(q, Q^2,:)\n",
    "\n",
    "    Wu * (uq - f(kappa, mu, xq[:,1], xq[:,2]))\n",
    "end\n",
    "\n",
    "function Compute_f1g0(kappa, mu, Q, num_comp, wdetJ, dXdx_T, q, Duq, pq)\n",
    "    \n",
    "    Wp = diagm(wdetJ)\n",
    "    Ws = diagm([wdetJ; wdetJ; wdetJ; wdetJ]);\n",
    "    \n",
    "    # sigma = -p I + 2 mu ed; where, ed = e - 1/3 * trace(e) * I, e = 0.5*(grad_u + grad_u^T)\n",
    "    s = zeros(2 * num_comp * Q^2)\n",
    "    p1  = zeros(Q^2)\n",
    "    for i=1:Q^2\n",
    "        grad_u1 = dXdx_T[i,:,:] * [Duq[i], Duq[i + Q^2]]\n",
    "        grad_u2 = dXdx_T[i,:,:] * [Duq[i + 2*Q^2], Duq[i + 3*Q^2]]\n",
    "        tr_e = grad_u1[1] + grad_u2[2]\n",
    "        # ed = e - 1/3 * trace(e) * I\n",
    "        ed11 = grad_u1[1] - 1/3. * tr_e\n",
    "        ed12 = (grad_u1[2] + grad_u2[1]) / 2.\n",
    "        ed22 = grad_u2[2] - 1/3. * tr_e\n",
    "        # -p I + 2 mu ed\n",
    "        s11 = 2 * mu * ed11 - pq[i]\n",
    "        s12 = 2 * mu * ed12\n",
    "        s22 = 2 * mu * ed22 - pq[i]\n",
    "        S1 = dXdx_T[i,:,:]' * vcat(s11', s12')\n",
    "        S2 = dXdx_T[i,:,:]' * vcat(s12', s22')\n",
    "        \n",
    "        s[i] = S1[1]\n",
    "        s[i + Q^2] = S1[2]\n",
    "        s[i + 2*Q^2] = S2[1]\n",
    "        s[i + 3*Q^2] = S2[2]\n",
    "        \n",
    "        # -(q, div(u)) -(q, p)/kappa\n",
    "        p1[i] = -tr_e - pq[i] / kappa\n",
    "        \n",
    "    end\n",
    "    # this is actually is f1, and g0\n",
    "    Ws * s, Wp * p1\n",
    "\n",
    "end\n",
    "\n",
    "# This is like libCEED/Ratel Residual Qfunction\n",
    "function Compute_df0(kappa, mu, Q, num_comp, wdetJ, q, uq, pq, du, dp, f)\n",
    "\n",
    "    Wu = diagm([wdetJ; wdetJ])\n",
    "    xq = reshape(q, Q^2,:)\n",
    "\n",
    "    Wu * du\n",
    "end\n",
    "\n",
    "function Compute_df1dg0(kappa, mu, Q, num_comp, wdetJ, dXdx_T, q, Duq, pq, Ddu, dp)\n",
    "    \n",
    "    Wp = diagm(wdetJ)\n",
    "    Ws = diagm([wdetJ; wdetJ; wdetJ; wdetJ]);\n",
    "    \n",
    "    ds = zeros(2 * num_comp * Q^2)\n",
    "    dp1  = zeros(Q^2)\n",
    "    for i=1:Q^2\n",
    "        grad_du1 = dXdx_T[i,:,:] * [Ddu[i], Ddu[i + Q^2]]\n",
    "        grad_du2 = dXdx_T[i,:,:] * [Ddu[i + 2*Q^2], Ddu[i + 3*Q^2]]\n",
    "        tr_de = grad_du1[1] + grad_du2[2]\n",
    "        # ded = de - 1/3 * trace(de) * I\n",
    "        ded11 = grad_du1[1] - 1/3. * tr_de\n",
    "        ded12 = (grad_du1[2] + grad_du2[1]) / 2.\n",
    "        ded22 = grad_du2[2] - 1/3. * tr_de\n",
    "        # -dp I + 2 mu ded\n",
    "        ds11 = 2 * mu * ded11 - dp[i]\n",
    "        ds12 = 2 * mu * ded12\n",
    "        ds22 = 2 * mu * ded22 - dp[i]\n",
    "        dS1 = dXdx_T[i,:,:]' * vcat(ds11', ds12')\n",
    "        dS2 = dXdx_T[i,:,:]' * vcat(ds12', ds22')\n",
    "        \n",
    "        ds[i] = dS1[1]\n",
    "        ds[i + Q^2] = dS1[2]\n",
    "        ds[i + 2*Q^2] = dS2[1]\n",
    "        ds[i + 3*Q^2] = dS2[2]\n",
    "        \n",
    "        # -(q, div(du)) -(q, dp)/kappa\n",
    "        dp1[i] = -tr_de - dp[i] / kappa\n",
    "        \n",
    "    end\n",
    "    \n",
    "    Ws * ds, Wp * dp1\n",
    "\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "c8e0d2c3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MixedElasticityJacobian (generic function with 1 method)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function MixedElasticityResidual(kappa, mu, u_in, xc, yc, Ind::FEIndices, Bx::FEBasis, Bu::FEBasis, f, bc_idx, u_bc)\n",
    "\n",
    "    u = copy(u_in); v = zero(u)\n",
    "    u[bc_idx] .= u_bc\n",
    "    \n",
    "    num_elem = size(Ind.idx_u,2)\n",
    "    \n",
    "    for e=1:num_elem\n",
    "        # Get coordinate of each element\n",
    "        Coord_E = GetCoordElem(xc, yc, Ind.Er_x, e);\n",
    "        # Update quadrature\n",
    "        q, wdetJ, dXdx_T = GetQdata(Coord_E, Bx);\n",
    "        \n",
    "        ue = Ind.Er_u[e] * u\n",
    "        pe = Ind.Er_p[e] * u\n",
    "        uq = 0 * Bu.B * ue\n",
    "        pq = Bu.Bp * pe\n",
    "        Duq = Bu.D * ue\n",
    "        \n",
    "        f0 = Compute_f0(kappa, mu, Bu.Q, Bu.num_comp, wdetJ, q, uq, pq, f);\n",
    "        f1, g0 = Compute_f1g0(kappa, mu, Bu.Q, Bu.num_comp, wdetJ, dXdx_T, q, Duq, pq);\n",
    "\n",
    "        ve = Bu.B' * f0 + Bu.D' * f1 # residual of the first equation (u)\n",
    "        qe = Bu.Bp' * g0             # residual of the first equation (p)\n",
    "        v += sparse(Ind.Er_u[e]') * ve + sparse(Ind.Er_p[e]') * qe\n",
    "    end   \n",
    "    v[bc_idx] = u_in[bc_idx] - u[bc_idx]\n",
    "    v\n",
    "\n",
    "end\n",
    "\n",
    "\n",
    "function MixedElasticityJacobian(kappa, mu, u_in, xc, yc, Ind::FEIndices, Bx::FEBasis, Bu::FEBasis, f, bc_idx, u_bc)\n",
    "\n",
    "    u = copy(u_in);\n",
    "    u[bc_idx] .= u_bc\n",
    "    \n",
    "    global_dof_u = Bu.num_comp * maximum(Ind.idx_u)\n",
    "    \n",
    "    num_elem = size(Ind.idx_u, 2)\n",
    "    rows, cols, vals = Int[], Int[], Float64[]\n",
    "    # size of the element\n",
    "    size_u = Bu.num_comp * Bu.P * Bu.P\n",
    "    if Ind.p_is_tensor\n",
    "        size_p = Bu.Pp * Bu.Pp\n",
    "    else\n",
    "        size_p = Bu.Pp\n",
    "    end\n",
    "    for e=1:num_elem\n",
    "        # Get coordinate of each element\n",
    "        Coord_E = GetCoordElem(xc, yc, Ind.Er_x, e);\n",
    "        # Update quadrature\n",
    "        q, wdetJ, dXdx_T = GetQdata(Coord_E, Bx);\n",
    "        ue = Ind.Er_u[e] * u\n",
    "        pe = Ind.Er_p[e] * u\n",
    "        Duq = Bu.D * ue\n",
    "        pq = Bu.Bp * pe\n",
    "        Jac = zeros(size_u + size_p, size_u + size_p);\n",
    "        for c=0:Bu.num_comp-1\n",
    "            for j=1:Bu.P * Bu.P\n",
    "                Ddu = Bu.D[:,j + c*Bu.P * Bu.P]\n",
    "                dp = 0 * Bu.Bp[:,1]\n",
    "                df1, dg0 = Compute_df1dg0(kappa, mu, Q, num_comp, wdetJ, dXdx_T, q, Duq, pq, Ddu, dp)\n",
    "                Jac[1:size_u,j + c*Bu.P * Bu.P] = Bu.D' * df1\n",
    "                Jac[size_u + 1:end,j + c*Bu.P * Bu.P] = Bu.Bp' * dg0\n",
    "            end\n",
    "        end\n",
    "        for j=1:size_p\n",
    "            Ddu = 0*Bu.D[:,1]\n",
    "            dp = Bu.Bp[:,j]\n",
    "            df1, dg0 = Compute_df1dg0(kappa, mu, Q, num_comp, wdetJ, dXdx_T, q, Duq, pq, Ddu, dp)\n",
    "            Jac[1:size_u, j + size_u] = Bu.D' * df1\n",
    "            Jac[size_u + 1:end, j + size_u] = Bu.Bp' * dg0\n",
    "        end\n",
    "        inds = rowvals(sparse(Ind.Er_u[e]'))\n",
    "        append!(inds, rowvals(sparse(Ind.Er_p[e]')))\n",
    "        append!(rows, kron(ones(size_u + size_p), inds))\n",
    "        append!(cols, kron(inds, ones(size_u + size_p)))\n",
    "        append!(vals, vec(Jac))\n",
    "    end   \n",
    "    A = sparse(rows, cols, vals)\n",
    "    A[bc_idx,:] .= 0.; A[:, bc_idx] .= 0.\n",
    "    for i=1:length(bc_idx)\n",
    "        A[bc_idx[i], bc_idx[i]] = 1.\n",
    "    end\n",
    "    A\n",
    "\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "dde4fa33",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(9.466582004640179e-5, 0.004167826125253)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "global_dof = maximum(Ind.idx_p)\n",
    "u0 = zeros(global_dof);\n",
    "\n",
    "# Dirichlet BCs\n",
    "bc_idx = GetDirichletBCsIndex(Bu.num_comp, Ind);\n",
    "x = GetNodalCoordinate(xc, yc, Bu.Q, Ind);\n",
    "x_bc = x[bc_idx]\n",
    "x_bc = reshape(x_bc,:,2)\n",
    "u_bc = uex(x_bc[:,1], x_bc[:,2])\n",
    "\n",
    "uh = nlsolve(u -> MixedElasticityResidual(kappa, mu, u, xc, yc, Ind, Bx, Bu, f, bc_idx, u_bc),\n",
    "             u -> MixedElasticityJacobian(kappa, mu, u, xc, yc, Ind, Bx, Bu, f, bc_idx, u_bc), u0; method = :newton)\n",
    "L2_error_u, L2_error_p = GetL2Error(kappa, xc, yc, Ind, Bx, Bu, uh.zero, uex, pex)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "92477b06",
   "metadata": {},
   "source": [
    "# <font color='blue'> Convergence study (Matrix-Free)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "88c29830",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(nu, E) = (0.375, 2.75)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(0.375, 2.75)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# P is number of nodes in 1D, P=2 means Q1 element\n",
    "# Pp is number of nodes in 1D if the pressure is tensor basis, otherwise is number of nodes in 2D\n",
    "P, Pp, Qmode, p_is_continuous, p_is_tensor = 4, 6, \"GAUSS\", false, false\n",
    "Q, num_comp = P, 2\n",
    "Bx = FEBasis(2, Pp, Q, 2, Qmode, p_is_tensor);\n",
    "Bu = FEBasis(P, Pp, Q, num_comp, Qmode, p_is_tensor);\n",
    "nx, ny, mesh, aspect_ratio= 2, 1, \"uniform\", 1\n",
    "xc, yc = GetCoordMesh(mesh, nx, ny, aspect_ratio);\n",
    "Ind = FEIndices(P, Pp, num_comp, nx, ny, p_is_continuous, p_is_tensor);\n",
    "lambda, mu = 3e0, 1;\n",
    "kappa = lambda + 2*mu/3;\n",
    "\n",
    "nu = 0.5*lambda/(lambda + mu)\n",
    "E = 2*mu*(1+nu)\n",
    "@show nu, E"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "ef7168cb",
   "metadata": {},
   "outputs": [],
   "source": [
    "res = [4, 8, 12, 16] # elements/side\n",
    "N = length(res)\n",
    "eu = zeros(N)\n",
    "ep = zeros(N)\n",
    "order_u = []\n",
    "append!(order_u, 0)\n",
    "order_p = []\n",
    "append!(order_p, 0)\n",
    "H = zeros(N)\n",
    "# refinement to plot convergence order\n",
    "for i=1:N\n",
    "\n",
    "    nx = res[i]\n",
    "    ny = res[i]\n",
    "\n",
    "    xc, yc = GetCoordMesh(mesh, nx, ny, aspect_ratio);\n",
    "    Ind = FEIndices(P, Pp, num_comp, nx, ny, p_is_continuous, p_is_tensor);\n",
    "    # Dirichlet BCs\n",
    "    bc_idx = GetDirichletBCsIndex(Bu.num_comp, Ind);\n",
    "    x = GetNodalCoordinate(xc, yc, Bu.Q, Ind);\n",
    "    x_bc = x[bc_idx]\n",
    "    x_bc = reshape(x_bc,:,2)\n",
    "    u_bc = uex(x_bc[:,1], x_bc[:,2])\n",
    "    \n",
    "    global_dof = maximum(Ind.idx_p)\n",
    "    u0 = zeros(global_dof);\n",
    "    Uh = nlsolve(u -> MixedElasticityResidual(kappa, mu, u, xc, yc, Ind, Bx, Bu, f, bc_idx, u_bc),\n",
    "                 u -> MixedElasticityJacobian(kappa, mu, u, xc, yc, Ind, Bx, Bu, f, bc_idx, u_bc), u0; method = :newton)\n",
    "    L2_error_u, L2_error_p = GetL2ErrorDisc(kappa, xc, yc, Ind, Bx, Bu, Uh.zero, uex, pex)\n",
    "    \n",
    "    eu[i] = L2_error_u\n",
    "    ep[i] = L2_error_p\n",
    "        \n",
    "    #h = Gethsz(xc, yc, IEN)\n",
    "    H[i] = 1/nx\n",
    "\n",
    "    if (i > 1)\n",
    "        logH = log10(H[i]/H[i-1])\n",
    "        append!(order_u, log10(eu[i]/eu[i-1])/logH)\n",
    "        append!(order_p, log10(ep[i]/ep[i-1])/logH)\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "46a437d9",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "***********************************************************************\n",
      "\n",
      "Convergence study (matrix-free) for 2D Mixed Elasticity for nu:0.375 with uniform mesh, number of nodes/elem u:4, p:6 \n",
      "\n",
      "Pressure field: (continuous, tensor)=(false, false)\n",
      "\n",
      "***********************************************************************\n",
      "\n",
      "\u001b[1m4×5 DataFrame\u001b[0m\n",
      "\u001b[1m Row \u001b[0m│\u001b[1m N     \u001b[0m\u001b[1m Error u    \u001b[0m\u001b[1m Order u \u001b[0m\u001b[1m Error p    \u001b[0m\u001b[1m Order p \u001b[0m\n",
      "\u001b[1m     \u001b[0m│\u001b[90m Int64 \u001b[0m\u001b[90m Float64    \u001b[0m\u001b[90m Any     \u001b[0m\u001b[90m Float64    \u001b[0m\u001b[90m Any     \u001b[0m\n",
      "─────┼─────────────────────────────────────────────────\n",
      "   1 │     4  2.41153e-5  0        0.00230616  0\n",
      "   2 │     8  1.59489e-6  3.91842  0.00030503  2.91847\n",
      "   3 │    12  3.19577e-7  3.96474  9.10127e-5  2.98277\n",
      "   4 │    16  1.01812e-7  3.97618  3.84514e-5  2.99498\n"
     ]
    }
   ],
   "source": [
    "# P is number of nodes/elem in 1D for u, convergence order should be \"P\"\n",
    "# Pp is number of nodes/elem in 1D for p, if pressure is tensor! otherwise is nodes/elem in 2D\n",
    "df = DataFrame(\"N\" => res, \"Error u\" => eu, \"Order u\" => order_u,\"Error p\" => ep, \"Order p\" => order_p)\n",
    "println(\"***********************************************************************\\n\")\n",
    "println(\"Convergence study (matrix-free) for 2D Mixed Elasticity for nu:$nu with $mesh mesh, number of nodes/elem u:$(P), p:$(Pp) \\n\")\n",
    "println(\"Pressure field: (continuous, tensor)=($p_is_continuous, $p_is_tensor)\\n\")\n",
    "println(\"***********************************************************************\\n\")\n",
    "println(df)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "ce75e183",
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plotmesh(mesh, xc, yc, Ind.idx_x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "50c0fb29",
   "metadata": {},
   "source": [
    "# <font color='blue'> Convergence study (Assembled)\n",
    " <font color='red'> It is slow, don't run it unless you need verification. See the solve time for the above mesh below for assmebled and matrix-free"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "bd16021f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "300.026718 seconds (435.10 k allocations: 385.339 GiB, 5.44% gc time)\n"
     ]
    }
   ],
   "source": [
    "@time GetFEMixed(kappa, mu, xc, yc, Ind, Bx, Bu, f, bc_idx, u_bc);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "168638d5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  2.302818 seconds (4.56 M allocations: 2.119 GiB, 7.36% gc time, 0.76% compilation time)\n"
     ]
    }
   ],
   "source": [
    "@time nlsolve(u -> MixedElasticityResidual(kappa, mu, u, xc, yc, Ind, Bx, Bu, f, bc_idx, u_bc),\n",
    "              u -> MixedElasticityJacobian(kappa, mu, u, xc, yc, Ind, Bx, Bu, f, bc_idx, u_bc), u0; method = :newton);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "b8d3011e",
   "metadata": {},
   "outputs": [],
   "source": [
    "res = [4, 8, 12,16]\n",
    "N = length(res)\n",
    "eu = zeros(N)\n",
    "ep = zeros(N)\n",
    "order_u = []\n",
    "append!(order_u, 0)\n",
    "order_p = []\n",
    "append!(order_p, 0)\n",
    "H = zeros(N)\n",
    "beta = zeros(N)\n",
    "# refinement to plot convergence order\n",
    "for i=1:N\n",
    "\n",
    "    nx = res[i]\n",
    "    ny = res[i]\n",
    "\n",
    "    xc, yc = GetCoordMesh(mesh, nx, ny, aspect_ratio);\n",
    "    Ind = FEIndices(P, Pp, num_comp, nx, ny, p_is_continuous, p_is_tensor);\n",
    "    # Dirichlet BCs\n",
    "    bc_idx = GetDirichletBCsIndex(Bu.num_comp, Ind);\n",
    "    x = GetNodalCoordinate(xc, yc, Bu.Q, Ind);\n",
    "    x_bc = x[bc_idx]\n",
    "    x_bc = reshape(x_bc,:,2)\n",
    "    u_bc = uex(x_bc[:,1], x_bc[:,2])\n",
    "    \n",
    "    Uh = GetFEMixed(kappa, mu, xc, yc, Ind, Bx, Bu, f, bc_idx, u_bc);\n",
    "    L2_error_u, L2_error_p = GetL2Error(kappa, xc, yc, Ind, Bx, Bu, Uh, uex, pex)\n",
    "    \n",
    "    eu[i] = L2_error_u\n",
    "    ep[i] = L2_error_p\n",
    "    \n",
    "    beta[i] = GetInfSup(xc, yc, Ind, Bx, Bu, \"\");\n",
    "    \n",
    "    #h = Gethsz(xc, yc, IEN)\n",
    "    H[i] = 1/nx\n",
    "\n",
    "    if (i > 1)\n",
    "        logH = log10(H[i]/H[i-1])\n",
    "        append!(order_u, log10(eu[i]/eu[i-1])/logH)\n",
    "        append!(order_p, log10(ep[i]/ep[i-1])/logH)\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "49583865",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "***********************************************************************\n",
      "\n",
      "Convergence study (assembled) for 2D Mixed Elasticity for nu:0.375 with uniform mesh, number of nodes/elem u:4, p:6 \n",
      "\n",
      "Pressure field: (continuous, tensor)=(false, false)\n",
      "\n",
      "***********************************************************************\n",
      "\n",
      "\u001b[1m4×6 DataFrame\u001b[0m\n",
      "\u001b[1m Row \u001b[0m│\u001b[1m N     \u001b[0m\u001b[1m Error u    \u001b[0m\u001b[1m Order u \u001b[0m\u001b[1m Error p     \u001b[0m\u001b[1m Order p \u001b[0m\u001b[1m Inf-sup constant \u001b[0m\n",
      "\u001b[1m     \u001b[0m│\u001b[90m Int64 \u001b[0m\u001b[90m Float64    \u001b[0m\u001b[90m Any     \u001b[0m\u001b[90m Float64     \u001b[0m\u001b[90m Any     \u001b[0m\u001b[90m Float64          \u001b[0m\n",
      "─────┼────────────────────────────────────────────────────────────────────\n",
      "   1 │     4  2.41153e-5  0        0.00146568   0                0.858153\n",
      "   2 │     8  1.59489e-6  3.91842  0.000194923  2.91059          0.821765\n",
      "   3 │    12  3.19577e-7  3.96474  5.83409e-5   2.97511          0.813919\n",
      "   4 │    16  1.01812e-7  3.97618  2.4688e-5    2.98936          0.81107\n"
     ]
    }
   ],
   "source": [
    "# P is number of nodes/elem in 1D for u, convergence order should be \"P\"\n",
    "# Pp is number of nodes/elem in 1D for p, if pressure is tensor! otherwise is nodes/elem in 2D\n",
    "df = DataFrame(\"N\" => res, \"Error u\" => eu, \"Order u\" => order_u,\"Error p\" => ep, \"Order p\" => order_p, \n",
    "\"Inf-sup constant\" => beta)\n",
    "println(\"***********************************************************************\\n\")\n",
    "println(\"Convergence study (assembled) for 2D Mixed Elasticity for nu:$nu with $mesh mesh, number of nodes/elem u:$(P), p:$(Pp) \\n\")\n",
    "println(\"Pressure field: (continuous, tensor)=($p_is_continuous, $p_is_tensor)\\n\")\n",
    "println(\"***********************************************************************\\n\")\n",
    "println(df)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7535d839",
   "metadata": {},
   "source": [
    "# Solve projection for P1 element defined on physical element"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "bbf75a72",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GetL2Error1 (generic function with 1 method)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function GetMassLocal1(wdetJ, Bp)\n",
    "    \n",
    "    W2 = diagm(wdetJ);\n",
    "    Me = Bp' * W2 * Bp\n",
    "    return Me\n",
    "end\n",
    "\n",
    "function GetRhsLocal1(kappa, wdetJ, q, Q, f, Bp)\n",
    "    \n",
    "    W2 = diagm(wdetJ);\n",
    "    xq = reshape(q, Q^2,:)\n",
    "    Fe = Bp' * W2 * f(kappa, xq[:,1], xq[:,2])  # v^T * f\n",
    "    return Fe\n",
    "end\n",
    "\n",
    "function elem_rest(idx_p)\n",
    "    \n",
    "    dof_p = maximum(idx_p)\n",
    "       \n",
    "    # element dof\n",
    "    elem_dof_p = size(idx_p, 1)\n",
    "    num_elem = size(idx_p, 2)\n",
    "    Er_p = []\n",
    "    for e = 1:num_elem\n",
    "        Lp = zeros(Int64, elem_dof_p, dof_p)\n",
    "        for i=1:elem_dof_p\n",
    "            Lp[i, idx_p[i, e] ] = 1\n",
    "        end\n",
    "            \n",
    "        push!(Er_p, Lp)\n",
    "    end\n",
    "    return Er_p\n",
    "end\n",
    "\n",
    "function GetFEProjection(kappa, xc, yc, Ind::FEIndices, Bx::FEBasis, idx_p, Er_p, f)\n",
    "    \n",
    "    num_elem = size(idx_p, 2)\n",
    "    global_dof = maximum(idx_p)\n",
    "    F = spzeros(global_dof, 1)\n",
    "    M = spzeros(global_dof, global_dof)\n",
    "    \n",
    "    for e=1:num_elem\n",
    "        # Get coordinate of each element\n",
    "        Coord_E = GetCoordElem(xc, yc, Ind.Er_x, e);\n",
    "        # Update quadrature\n",
    "        q, wdetJ, _ = GetQdata(Coord_E, Bx);\n",
    "        Bp = P1_basis_phy(q, Bx.Q)\n",
    "        Me = GetMassLocal1(wdetJ, Bp);  # v^T * u\n",
    "        Fe = GetRhsLocal1(kappa, wdetJ, q, Bx.Q, f, Bp);  # v^T * f\n",
    "\n",
    "        # Assemble\n",
    "        M = M + Er_p[e]' * Me * Er_p[e]\n",
    "        F = F + Er_p[e]' * Fe\n",
    "\n",
    "    end\n",
    "    u = M \\ F\n",
    "    \n",
    "   return u \n",
    "end\n",
    "\n",
    "function GetL2Error1(kappa, xc, yc, Ind::FEIndices, Bx::FEBasis, idx_p, Er_p, ph, pex)\n",
    "    \"\"\"This function returns L2 error \n",
    "    (sum_{e=1:num_elem} (int_e (uh - uex)^2 dx) )^0.5 \n",
    "    \"\"\"\n",
    "\n",
    "    num_elem = size(Ind.idx_u,2)\n",
    "\n",
    "    e_p = zeros(num_elem)\n",
    "    for e=1:num_elem\n",
    "        # Get coordinate of each element\n",
    "        Coord_E = GetCoordElem(xc, yc, Ind.Er_x, e);\n",
    "        # Update quadrature\n",
    "        q, wdetJ, _ = GetQdata(Coord_E, Bx);\n",
    "        Bp = P1_basis_phy(q, Bx.Q)\n",
    "        phh = Er_p[e] * ph\n",
    "        p_num = Bp * phh\n",
    "        \n",
    "        xq = reshape(q, Q^2,:)\n",
    "        p_ex = pex(kappa, xq[:,1], xq[:,2])\n",
    "\n",
    "        err_p = vec((p_num - p_ex))\n",
    "        # \\int (u_num - u_ex)^2 dx ==> on element e\n",
    "        e_p[e] = wdetJ' * (err_p .* err_p)\n",
    "\n",
    "    end\n",
    "\n",
    "    return sqrt(sum(e_p))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "a95e1fa7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Any}:\n",
       " 0\n",
       " 1.962610547454403\n",
       " 1.988186180140432\n",
       " 1.994174649443653"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# P is number of nodes in 1D, P=2 means Q1 element\n",
    "# Pp is number of nodes in 1D if the pressure is tensor basis, otherwise is number of nodes in 2D\n",
    "P, Pp, Qmode, p_is_continuous, p_is_tensor = 3, 3, \"GAUSS\", false, false\n",
    "Q, num_comp = P, 2\n",
    "Bx = FEBasis(2, Pp, Q, 2, Qmode, p_is_tensor);\n",
    "nx, ny, mesh, aspect_ratio= 2,1, \"uniform\", 10\n",
    "xc, yc = GetCoordMesh(mesh, nx, ny, aspect_ratio);\n",
    "Ind = FEIndices(P, Pp, num_comp, nx, ny, p_is_continuous, p_is_tensor);\n",
    "lambda, mu = 3, 1;\n",
    "kappa = lambda + 2*mu/3;\n",
    "\n",
    "idx_p = GetConnectivityPressure(3, nx, ny, false, false);\n",
    "Er_p = elem_rest(idx_p)\n",
    "ph = GetFEProjection(kappa, xc, yc, Ind, Bx, idx_p, Er_p, pex);\n",
    "GetL2Error1(kappa, xc, yc, Ind, Bx, idx_p, Er_p, ph, pex);\n",
    "\n",
    "res = [4, 8, 12, 16] # elements/side\n",
    "N = length(res)\n",
    "ep = zeros(N)\n",
    "order_p = []\n",
    "append!(order_p, 0)\n",
    "H = zeros(N)\n",
    "# refinement to plot convergence order\n",
    "for i=1:N\n",
    "\n",
    "    nx = res[i]\n",
    "    ny = res[i]\n",
    "\n",
    "    xc, yc = GetCoordMesh(mesh, nx, ny, aspect_ratio);\n",
    "    Ind = FEIndices(P, Pp, num_comp, nx, ny, p_is_continuous, p_is_tensor);\n",
    "    idx_p = GetConnectivityPressure(3, nx, ny, false, false);\n",
    "    Er_p = elem_rest(idx_p)\n",
    "    ph = GetFEProjection(kappa, xc, yc, Ind, Bx, idx_p, Er_p, pex);\n",
    "    eee = GetL2Error1(kappa, xc, yc, Ind, Bx, idx_p, Er_p, ph, pex);\n",
    "    \n",
    "    ep[i] = eee\n",
    "        \n",
    "    #h = Gethsz(xc, yc, IEN)\n",
    "    H[i] = 1/nx\n",
    "\n",
    "    if (i > 1)\n",
    "        logH = log10(H[i]/H[i-1])\n",
    "        append!(order_p, log10(ep[i]/ep[i-1])/logH)\n",
    "    end\n",
    "end\n",
    "order_p"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "00cec702",
   "metadata": {},
   "source": [
    "# Inf-Sup results for discontinuous pressure and element aspect ratio 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "3ada66ac",
   "metadata": {},
   "outputs": [],
   "source": [
    "label_dis = [\"Q1P0\" \"Q2P0\" \"Q2P1-ref\" \"Q2P1-phy\" \"Q3Q1\" \"Q3Q2\" \"Q4Q1\" \"Q4Q2\"]\n",
    "# Set: (u, p) for above elements\n",
    "order_dis = [2 1; 3 1; 3 3;3 3; 4 2; 4 3; 5 2; 5 3]\n",
    "p_is_tensors = [false; false; false; false; true; true; true; true]\n",
    "mesh, aspect_ratio= \"uniform\", 1\n",
    "num_comp, Qmode, p_is_continuous = 2, \"GAUSS\", false\n",
    "\n",
    "res = 2:8\n",
    "N = length(res)\n",
    "H = zeros(N)\n",
    "M = length(label_dis)\n",
    "beta_dis_1 = zeros(N, M)\n",
    "\n",
    "for j=1:M\n",
    "\n",
    "    P, Pp, p_is_tensor = order_dis[j,1], order_dis[j,2], p_is_tensors[j,1]\n",
    "    Q = P\n",
    "    Bx = FEBasis(2, Pp, Q, 2, Qmode, p_is_tensor);\n",
    "    Bu = FEBasis(P, Pp, Q, num_comp, Qmode, p_is_tensor);\n",
    "    if j==4\n",
    "        P1_config = \"physical\"\n",
    "    else\n",
    "        P1_config = \"\"\n",
    "    end\n",
    "\n",
    "    # refinement to plot convergence order\n",
    "    for i=1:N\n",
    "\n",
    "        nx = res[i]\n",
    "        ny = res[i]\n",
    "\n",
    "        xc, yc = GetCoordMesh(mesh, nx, ny, aspect_ratio);\n",
    "        Ind = FEIndices(P, Pp, num_comp, nx, ny, p_is_continuous, p_is_tensor);\n",
    "        beta_dis_1[i, j] = GetInfSup(xc, yc, Ind, Bx, Bu, P1_config);\n",
    "    \n",
    "        #h = Gethsz(xc, yc, IEN)\n",
    "        H[i] = 1/nx\n",
    "\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "4d2d6b91",
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = plot(title=\"Discontinuous Pressure, Element aspect ratio 1\", xlabel=\"h\", ylabel=\"inf-sup constant\",\n",
    "    legend=:bottomright, legend_font=12)\n",
    "for j=1:M\n",
    "    plot!(p, H, beta_dis_1[:,j], xaxis=:log, lw = 3,ylims=(0, 1.2), label = label_dis[j])\n",
    "end\n",
    "p"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6f9ddeed",
   "metadata": {},
   "source": [
    "# Inf-Sup results for discontinuous pressure and element aspect ratio 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "c9e623b3",
   "metadata": {},
   "outputs": [],
   "source": [
    "label_dis = [\"Q1P0\" \"Q2P0\" \"Q2P1-ref\" \"Q2P1-phy\" \"Q3Q1\" \"Q3Q2\" \"Q4Q1\" \"Q4Q2\"]\n",
    "# Set: (u, p) for above elements\n",
    "order_dis = [2 1; 3 1; 3 3; 3 3; 4 2; 4 3; 5 2; 5 3]\n",
    "p_is_tensors = [false; false; false; false; true; true; true; true]\n",
    "mesh, aspect_ratio= \"uniform\", 10\n",
    "num_comp, Qmode, p_is_continuous = 2, \"GAUSS\", false\n",
    "\n",
    "res = 2:8\n",
    "N = length(res)\n",
    "H = zeros(N)\n",
    "M = length(label_dis)\n",
    "beta_dis_10 = zeros(N, M)\n",
    "\n",
    "for j=1:M\n",
    "\n",
    "    P, Pp, p_is_tensor = order_dis[j,1], order_dis[j,2], p_is_tensors[j,1]\n",
    "    Q = P\n",
    "    Bx = FEBasis(2, Pp, Q, 2, Qmode, p_is_tensor);\n",
    "    Bu = FEBasis(P, Pp, Q, num_comp, Qmode, p_is_tensor);\n",
    "    if j==4\n",
    "        P1_config = \"physical\"\n",
    "    else\n",
    "        P1_config = \"\"\n",
    "    end\n",
    "\n",
    "    # refinement to plot convergence order\n",
    "    for i=1:N\n",
    "\n",
    "        nx = res[i]\n",
    "        ny = res[i]\n",
    "\n",
    "        xc, yc = GetCoordMesh(mesh, nx, ny, aspect_ratio);\n",
    "        Ind = FEIndices(P, Pp, num_comp, nx, ny, p_is_continuous, p_is_tensor);\n",
    "    \n",
    "        beta_dis_10[i, j] = GetInfSup(xc, yc, Ind, Bx, Bu, P1_config)\n",
    "    \n",
    "        #h = Gethsz(xc, yc, IEN)\n",
    "        H[i] = 1/nx\n",
    "\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "bf38c704",
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = plot(title=\"Discontinuous Pressure, Element aspect ratio 10\", xlabel=\"h\", ylabel=\"inf-sup constant\",\n",
    "    legend=:bottomright, legend_font=12)\n",
    "for j=1:M\n",
    "    plot!(p, H, beta_dis_10[:,j], xaxis=:log, lw = 3,ylims=(0, 1.2), label = label_dis[j])\n",
    "end\n",
    "p"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3e02ee8e",
   "metadata": {},
   "source": [
    "# Inf-Sup results for continuous pressure and element aspect ratio 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "e3e75803",
   "metadata": {},
   "outputs": [],
   "source": [
    "label_cont = [\"Q2Q1\" \"Q3Q1\" \"Q3Q2\" \"Q4Q1\" \"Q4Q2\" \"Q4Q3\"]\n",
    "order_cont = [3 2; 4 2; 4 3; 5 2; 5 3; 5 4]\n",
    "mesh, aspect_ratio= \"uniform\", 1\n",
    "num_comp, Qmode, p_is_continuous, p_is_tensor = 2, \"GAUSS\", true, true\n",
    "\n",
    "res = 2:8\n",
    "N = length(res)\n",
    "H = zeros(N)\n",
    "M = length(label_cont)\n",
    "beta_cont_1 = zeros(N, M)\n",
    "\n",
    "for j=1:M\n",
    "\n",
    "P, Pp = order_cont[j,1], order_cont[j,2]\n",
    "Q = P\n",
    "Bx = FEBasis(2, Pp, Q, 2, Qmode, p_is_tensor);\n",
    "Bu = FEBasis(P, Pp, Q, num_comp, Qmode, p_is_tensor);\n",
    "\n",
    "    # refinement to plot convergence order\n",
    "    for i=1:N\n",
    "\n",
    "        nx = res[i]\n",
    "        ny = res[i]\n",
    "\n",
    "        xc, yc = GetCoordMesh(mesh, nx, ny, aspect_ratio);\n",
    "        Ind = FEIndices(P, Pp, num_comp, nx, ny, p_is_continuous, p_is_tensor);\n",
    "    \n",
    "        beta_cont_1[i, j] = GetInfSup(xc, yc, Ind, Bx, Bu,\"\")\n",
    "    \n",
    "        #h = Gethsz(xc, yc, IEN)\n",
    "        H[i] = 1/nx\n",
    "\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "a9c0ddbd",
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = plot(title=\"Continuous Pressure, Element aspect ratio 1\", xlabel=\"h\", ylabel=\"inf-sup constant\",\n",
    "    legend=:bottomright, legend_font=12)\n",
    "for j=1:M\n",
    "    plot!(p, H, beta_cont_1[:,j], xaxis=:log, lw = 3,ylims=(0, 1.2), label = label_cont[j])\n",
    "end\n",
    "p"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1f8609b5",
   "metadata": {},
   "source": [
    "# Inf-Sup results for continuous pressure and element aspect ratio 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "632aa933",
   "metadata": {},
   "outputs": [],
   "source": [
    "label_cont = [\"Q2Q1\" \"Q3Q1\" \"Q3Q2\" \"Q4Q1\" \"Q4Q2\" \"Q4Q3\"]\n",
    "order_cont = [3 2; 4 2; 4 3; 5 2; 5 3; 5 4]\n",
    "mesh, aspect_ratio= \"uniform\", 10\n",
    "num_comp, Qmode, p_is_continuous, p_is_tensor = 2, \"GAUSS\", true, true\n",
    "\n",
    "res = 2:8\n",
    "N = length(res)\n",
    "H = zeros(N)\n",
    "M = length(label_cont)\n",
    "beta_cont_10 = zeros(N, M)\n",
    "\n",
    "for j=1:M\n",
    "\n",
    "P, Pp = order_cont[j,1], order_cont[j,2]\n",
    "Q = P\n",
    "Bx = FEBasis(2, Pp, Q, 2, Qmode, p_is_tensor);\n",
    "Bu = FEBasis(P, Pp, Q, num_comp, Qmode, p_is_tensor);\n",
    "\n",
    "    # refinement to plot convergence order\n",
    "    for i=1:N\n",
    "\n",
    "        nx = res[i]\n",
    "        ny = res[i]\n",
    "\n",
    "        xc, yc = GetCoordMesh(mesh, nx, ny, aspect_ratio);\n",
    "        Ind = FEIndices(P, Pp, num_comp, nx, ny, p_is_continuous, p_is_tensor);\n",
    "    \n",
    "        beta_cont_10[i, j] = GetInfSup(xc, yc, Ind, Bx, Bu, \"\")\n",
    "    \n",
    "        #h = Gethsz(xc, yc, IEN)\n",
    "        H[i] = 1/nx\n",
    "\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "8d91e09c",
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = plot(title=\"Continuous Pressure, Element aspect ratio 10\", xlabel=\"h\", ylabel=\"inf-sup constant\",\n",
    "    legend=:bottomright, legend_font=12)\n",
    "for j=1:M\n",
    "    plot!(p, H, beta_cont_10[:,j], xaxis=:log, lw = 3,ylims=(0, 1.2), label = label_cont[j])\n",
    "end\n",
    "p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1bb17e8a",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3f566a9c",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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