Groundwater Flow (Robin)

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Equations

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 \[ \eqalign{ h(x) = g_D(x) &\quad \text{on }\Gamma_D,\cr k{\partial h(x) \over \partial n} = g_N(x) &\quad \text{on }\Gamma_N, }\] 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 \[ \begin{equation} h(x,y) = 1 + \sum_{k=1}^\infty A_k \sin\bigg(C_k y\bigg) \sinh\bigg(C_k x\bigg), \end{equation} \]


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