NumGy_LV12E1.qmd

---
title: "Numerical Simulation Methods in Geophysics, Part 6: Finite Elements"
subtitle: "1. MGPY+MGIN"
author: "thomas.guenther@geophysik.tu-freiberg.de"
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--- 
## Variational formulation of Diffusion equation

$$ \pdv{u}{t} - \div a \grad u = f $$

::: {.fragment}
Multiplication with test function $w$ and integration $\Rightarrow$ weak form

$$ -\int_\Omega w \div a \grad u \dd\Omega = \int_\Omega w f \dd\Omega $$
:::

::: {.fragment}
$$ \div(b\vb c) = b\div \vb c + \grad b \cdot \vb c $$
:::

::: {.fragment}
$$ \int_\Omega a \grad w \cdot \grad u \dd\Omega - \int_\Omega \div(w a \grad u) \dd\Omega = \int_\Omega w f \dd\Omega $$
:::

## Variational formulation of Poisson equation

$$ \int_\Omega a \grad w \cdot \grad u \dd\Omega - \int_\Omega \div(w a \grad u) \dd\Omega = \int_\Omega w f \dd\Omega $$

::: {.fragment}
use Gauss' law $\int_\Omega \div \vb A = \int_\Gamma \vb A \cdot \vb n$

$$ \int_\Omega a \grad w \cdot \grad u \dd\Omega - \int_\Gamma a w \grad u \cdot \vb n \dd\Gamma = \int_\Omega fw \dd\Omega $$
:::

::: {.fragment}
Let $u$ be constructed by shape functions $v$: $u=\sum_i u_i v_i$

$$ \int_\Omega a \grad w \cdot \grad v_i \dd\Omega - \int_\Gamma a w \grad v_i \cdot \vb n \dd\Gamma = \int_\Omega fw \dd\Omega $$
:::

## Galerkin's method

$$ \int_\Omega a \grad w \vdot \grad v_i \dd\Omega - \int_\Gamma a w \grad v_i \cdot \vb n \dd\Gamma = \int_\Omega fw \dd\Omega $$

::: {.fragment}
Test functions the same as shape (trial) functions $w\in  v_i$

$$ \int_\Omega a \grad v_j \vdot \grad v_i \dd\Omega - \int_\Gamma a v_j \grad v_i \vdot \vb n \dd\Gamma = \int_\Omega f v_j \dd\Omega $$
:::

::: {.fragment}
* choose $v_i$ so that $\grad v_i$ is simple and $\grad v_i \vdot \grad v_j$ mostly 0
* divide subsurface in sub-volumes $\Omega_i$ with constant $a_i$ (& $\grad v_j$)
:::

## Hat functions 

![](pics/shapefunctions1d.svg){width="100%"}

## Gradients for hat functions
Every element is surrounded by two nodes "carrying" a hat. 
The gradients are piece-wise constant $\pm 1/\Delta x_i$

Neighboring functions $v_i$ & $v_{i+1}$ only meet between $x_i$ & $x_{i+1}$

$$ \int_\Omega a \grad v_i \vdot \grad v_{i+1} \dd\Omega = \int_{x_i}^{x_{i+1}} a \grad v_i \vdot \grad v_{i+1} \dd\Omega = -\frac{a_i}{\Delta x_i^2} \Delta x_i = -\frac{a_i}{\Delta x_i}$$ 

$$ -\int_\Omega a \grad v_i \vdot \grad v_i \dd\Omega = \frac{a_{i-1}}{\Delta x_{i-1}^2}\Delta x_{i-1} + \frac{a_i}{\Delta x_i^2}\Delta x_i = \frac{a_{i-1}}{\Delta x_{i-1}} + \frac{a_i}{\Delta x_i} $$

<!-- $$ -\int_\Omega a \grad v_i \vdot \grad v_{i+2} \dd\Omega = 0 $$ -->

## Integration

Let's write it up for the first 

$$ \int u_0 a v_0' v_0' + \int u_1 a v_0' v_1' = \int v_0 f $$

$$ \int u_0 a v_0' v_1' + \int u_1 a v_1' v_1' + \int a u_2 v_2' v_1' = \int v_1 f $$

<!-- $$ \int u_1 v_1' v_2' + \int u_2 v_2' v_2' + \int u_3 v_3' v_2' = \int v_1 f $$ -->

$$ u_{i-1} a_{i-1} \int\limits_{x_{i-1}}^{x_{i}} v_i' v_{i-1}' + 
   u_i a_{i-1} \int\limits_{x_{i-1}}^{x_{i}} v_i' v_i' + 
   u_i a_i \int\limits_{x_{i}}^{x_{i+1}} v_i' v_i' + 
   u_{i+1} a_{i} \int\limits_{x_{i}}^{x_{i+1}} v_i' v_{i+1}' $$


# System (stiffness) matrix

Matrix integrating gradient of base functions for neighbors
$A$ 
$$ \vb A_{i, i+1} = -\frac{a_i}{\Delta x_i^2} \vdot \Delta x_i = -\frac{a_i}{\Delta x_i} $$

$$ A_{i,i} = \int_\Omega a \grad v_i \cdot \grad v_i \dd\Omega = -A_{i,i+1} - A_{i+1,i}  $$

$\Rightarrow$ matrix-vector equation $\vb A \vb u = \vb b$ with conductances in $\vb A$
<!-- $\Rightarrow$ -->

## Boundary conditions

second term
$$ - \int_\Gamma a v_j \grad v_i \cdot \vb n \dd\Gamma $$

$$ [a v_i v_j']_{x_0}^{x_N} = a_{N-1} u_N v'_N - a_0 u_0 v'_0 $$

Homogeneous Neumann automatically 

## Right-hand side vector

The right-hand-side vector $b=\int v_i f \dd\Omega$ also scales with $\Delta x$

e.g. $f=\div \vb j_s$ $\Rightarrow$ 
$b=\int v_i \div \vb j_s \dd\Omega = \int_\Gamma v_i \vb j_S \cdot \vb n$

system identical to FD for $\Delta x=$const

:::{.callout-tip icon="false" title="Difference of FE to FD"} 
Any source function $f(x)$ can be integrated on the whole space!
:::

## Solution

$\vb u$ holds the coefficient $u_i$ creating $u(x)=\sum u_i v_i(x)$

:::{.callout-tip icon="false" title="Difference of FE to FD"} 
$u$ is described on the whole space and approximates the solution, not the PDE!
:::

Hat functions: $u_i$ potentials on nodes, $u$ piece-wise linear

:::{.callout-tip icon="false" title="Generality of FE"} 
Arbitrary base functions $v_i$ can be used to describe $u$
:::