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EDIT: I am trying to specify my Question. Also I am not going to clearify which spaces I use, because I am only interested in the basic idea.

I am looking at a standard elliptic second order PDE:

\begin{cases} Lu & = f & \text{in} &\Omega \\ \quad u & = g & \text{on} & \Gamma \subset \partial\Omega \end{cases}

where $\Omega \subset \mathbb{R}^n$ is an open bounded Lipschitz domain. My goal is to find the normal derivative $\partial_n u$ on the boundary $\Gamma$ using the Dirichlet-to-Neumann (DtN) Operator in the Framework of FEM.

As far as I know the DtN operator is as map

\begin{equation} \Lambda_\Gamma:g \to\partial_nu \quad with \quad u=0 \quad on\quad \partial\Omega\setminus\Gamma \end{equation}

where $u$ is the solution of the boundary value problem.

For $Lu = - \Delta u$ the weak form of the pde is given as: Find $u$ such that

\begin{equation} \int_\Omega \nabla u\nabla v \,\,d\Omega = \int_\Omega f v \,\, d\Omega \qquad\forall v \end{equation} with $u = g$ on $\Gamma$.

My question is, how do I use the DtN operator to calculate the normal derivative in this case?

1 Answers1

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The DtN operator has no closed form expression on general domains. Rather, the way you evaluate it is that you need to solve the Laplace equation $-\Delta u = 0$ with given boundary conditions $u|_{\partial\Omega}=g$ for the function $u$ and then take its normal derivative at the boundary as output. Of course, in general, we can't solve the Laplace equation either, so you have to use a numerical approximation $u_h$ of $u$, and then take that function's normal derivative at the boundary instead.

Wolfgang Bangerth
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  • I have been told that the DtN map can be discretized/approximated by Schur complementing the interior of $\Omega$ to the boundary $\partial\Omega$, but I have never sought a reference/derivation of this. Is that something that would be worth including in your answer? – rchilton1980 May 17 '19 at 14:26
  • After discretization by, say the finite element method, you compute $u_h=\sum_j U_j \varphi_j$ through solving the linear system $AU=HG$ where $HG$ contains the effects of the Dirichlet boundary conditions. Here, $G$ is the set of boundary values at boundary points, and $H$ is the effect of these when multiplied by appropriate shape functions; $H$ is a rectangular matrix. So $U=A^{-1}HG$. Then you have to compute the Neumann values, which requires applying a rectangular matrix $B$ to the set of coefficients $U$. This means that the DtN operator is $BA^{-1}H$. – Wolfgang Bangerth May 17 '19 at 20:42