Project file on GitHub
## Equations

## First benchmark: Problem specification

### Analytical solution

### Results and evaluation

## Second benchmark: Problem specification and analytical solution

We start with simple linear homogeneous elliptic problem: \[ \begin{equation*} k\; \Delta h = 0 \quad \text{in }\Omega \end{equation*}\] w.r.t boundary conditions \[ \eqalign{ h = g_D &\quad \text{on }\Gamma_D,\cr k{\partial h \over \partial n} = g_N &\quad \text{on }\Gamma_N,\cr {\partial h \over \partial n} = \alpha (h_0 - h(x)) &\quad \text{on }\Gamma_R, }\] where \(h\) could be hydraulic head, pressure, or temperature and \(k\) is the diffusion tensor (hydraulic conductivity, permeability divided by dynamic viscosity, or heat conductivity). The subscripts \(D,\) \(N,\) and \(R\) denote the Dirichlet-, Neumann-, and Robin-type boundary conditions, \(n\) is the normal vector pointing outside of \(\Omega\), and \(\Gamma = \Gamma_D \cup \Gamma_N \cup \Gamma_R\) and \(\Gamma_D \cap \Gamma_N \cap \Gamma_R = \emptyset\).

We solve the Laplace equation on a line domain \([0, 1]\) with \(k = 1\) w.r.t. the specific boundary conditions: \[
\eqalign{
{\partial h \over \partial n} = \alpha (h_0 - h(x)) &\quad \text{for } x=0,\cr
h(x) = g_D &\quad \text{for } x=1,
}\] see `line_1e1_robin_left_picard.prj`

.

One particular solution is \[ \begin{equation*} h(x) = A x + B. \end{equation*} \]

The normal direction is facing out of the bulk domain. The Robin-type boundary condition in this example is set on the left side of the line domain. Consequently, in this case the directional derivative is the negative derivative \[ \begin{equation*} \left.\frac{\partial h}{\partial n}\right\rvert_{x=0} = -h'(x)|_{x=0}. \end{equation*} \] From the evaluation of the the Robin-type boundary condition it follows \[ \begin{equation*} \left.\frac{\partial h}{\partial n}\right\rvert_{x=0} = -A = \alpha (h_0 - h(0)) = \alpha (h_0 - B). \end{equation*} \] Using the expression for \(A\) in the Dirichlet-type boundary condition \[ \begin{equation*} h(x)|_{x=1} = A + B = -\alpha (h_0 - B) + B = -\alpha h_0 + (1+\alpha) B = g_D \end{equation*} \] yields for \(\alpha \not= -1\): \[ \begin{align*} B &= \frac{g_D + \alpha h_0}{1 + \alpha} \quad \textrm{and}\\ A &= -\alpha \left( h_0 - \frac{g_D + \alpha h_0}{1 + \alpha} \right) = -\alpha \left( \frac{h_0 + \alpha h_0 - g_D - \alpha h_0}{1 + \alpha} \right) = -\alpha \left( \frac{h_0 - g_D}{1 + \alpha} \right). \end{align*} \] The particular solution is \[ \begin{equation*} h(x) = \frac{\alpha (g_D - h_0)}{1 + \alpha} x + \frac{g_D + \alpha h_0}{1 + \alpha}. \end{equation*} \] Using the values from the project file \(\alpha = -2,\) \(h_0 = 1.5\), \(g_D = 2\) results in \[ \begin{equation*} h(x) = \frac{-2 (2 - 1.5)}{1 + (-2)} x + \frac{2+(-2) \times 1.5}{1 + (-2)} = x + 1. \end{equation*} \]

The left figure shows the pressure along the line, in the right figure the difference between the analytical solution and the numerical calculated solution is plotted.

We solve the Laplace equation on a line domain \([0, 1]\) with \(k = 1\) w.r.t. the specific boundary conditions: \[
\eqalign{
h(x) = g_D &\quad \text{for } x=0,\cr
{\partial h \over \partial n} = \alpha (h_0 - h(x)) &\quad \text{for } x=1,
}\] see `line_1e1_robin_right_picard.prj`

.

One particular solution is \[ \begin{equation*} h(x) = A x + B. \end{equation*} \] Due to the Dirichlet boundary condition it follows: \[ \begin{equation*} h(0) = g_D = B \quad \Rightarrow \quad h(x) = A x + g_D. \end{equation*} \] From the Robin-type boundary condition we get \[ \begin{equation*} h'(x)|_{x=1} = A = \alpha \left(h_0 - h(x)|_{x=1} \right) = \alpha \left.\left(h_0 - (Ax+g_D)\right)\right\rvert_{x=1} = \alpha (h_0 - g_D) - \alpha A. \end{equation*} \] \[ \begin{equation*} \Rightarrow A = \frac{\alpha (h_0 - g_D)}{1+\alpha} \end{equation*} \] \[ \begin{equation*} h(x) = \frac{\alpha (h_0 - g_D)}{1+\alpha} x + g_D. \end{equation*} \] The values from the project file are: \(\alpha = -2,\) \(h_0 = 1.5\), \(g_D = 1\) yielding \[ \begin{equation*} h(x) = x + 1. \end{equation*} \]

This article was written by Thomas Fischer, Dmitri Naumov. If you are missing something or you find an error please let us know.
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