groundwater-flow-dirichlet-nodal-source-term.pandoc

+++
date = "2017-02-15T11:17:39+01:00"
title = "Groundwater Flow (Nodal Source Term)"
project = "Elliptic/circle_radius_1/circle_1e6_axi.prj"
author = "Thomas Fischer"
weight = 104

[menu]
  [menu.benchmarks]
    parent = "elliptic"

+++

{{< data-link >}}

## Equations

We solve the Poisson equation:
$$
\begin{equation}
k\; \Delta h = f(x) \quad \text{in }\Omega
\end{equation}$$
w.r.t boundary conditions
$$
\eqalign{
h(x) = g_D(x) &\quad \text{on }\Gamma_D,\cr
k\;{\partial h(x) \over \partial n} = g_N(x) &\quad \text{on }\Gamma_N,
}$$

where $h$ could be hydraulic head, the subscripts $D$ and $N$ denote the Dirichlet- and Neumann-type boundary conditions, $n$ is the normal vector pointing outside of $\Omega$, and $\Gamma = \Gamma_D \cup \Gamma_N$ and $\Gamma_D \cap \Gamma_N = \emptyset$.

## Problem specification and analytical solution

We solve the Poisson equation on a circle domain with radius $r = 1$ with $k = 1$ w.r.t. the specific boundary conditions:
$$
\eqalign{
h(x,y) = 0 &\quad \text{on } (x^2 + y^2 = 1) \subset \Gamma_D,\cr
}$$
The solution of this problem is
$$
h(x,y) = \int \int f(\xi, \eta) G(x, y) d \xi d \eta,
$$
where $G(x, y)$ is the Green's function. For the example at hand $G(x, y)$ is:
$$
G(x, y) = \frac{1}{2 \pi} \ln \sqrt{(x-\xi)^2 + (y-\eta)^2}.
$$
With a nodal source term of 1 at $(0.0, 0.0)$ the analytical solution is
$$
h(x,y) = \frac{1}{2 \pi} \ln \sqrt{x^2 + y^2}.
$$


## Input files

The main project file is `square_1e6_with_nodal_sources.prj`. It describes the process to be solved and the related process variables together with their initial and boundary conditions as well as the definition of the nodal source term. It also references the mesh and geometrical objects defined on the mesh.

The geometries used to specify the boundary conditions and the source term are given in the `square_1x1.gml` file.

The input mesh `square_1x1_quad_1e6.vtu` is stored in the VTK file format and can be directly visualized in Paraview for example.

## Running simulation

To start the simulation (after successful compilation) run:
```bash
$ ogs circle_1e6_axi.prj
```

It will produce some output and write the computed result into a data array of the written vtu file.


## Results and evaluation

### Comparison of the analytical solution and the computed solution

{{< img src="../circle_1e6_gwf_with_nodal_source_term_analytical_solution_head.png" >}}

{{< img src="../circle_1e6_gwf_with_nodal_source_term_diff_analytical_solution_head.png" >}}

{{< img src="../circle_1e6_gwf_with_nodal_source_term_diff_analytical_solution_head_log_scale.png" >}}