# Groundwater Flow (Robin)

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## Equations

We start with simple linear homogeneous elliptic problem: $$$k\; \Delta h = 0 \quad \text{in }\Omega$$$ 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 Laplace equation on a line domain $[0\times 1]^2$ with $k = 1$ w.r.t. the specific boundary conditions:

\eqalign{ h(x,y) = 1 &\quad \text{on } (x=0,y) \subset \Gamma_D,\cr h(x,y) = 1 &\quad \text{on } (x,y=0) \subset \Gamma_D,\cr k {\partial h(x,y) \over \partial n} = 1 &\quad \text{on } (x=1,y) \subset \Gamma_N,\cr k {\partial h(x,y) \over \partial n} = 0 &\quad \text{on } (x,y=1) \subset \Gamma_N. }

The solution of this problem is $$$h(x,y) = 1 + \sum_{k=1}^\infty A_k \sin\bigg(C_k y\bigg) \sinh\bigg(C_k x\bigg),$$$

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