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?


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.

  • $\begingroup$ 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? $\endgroup$ May 17 '19 at 14:26
  • $\begingroup$ 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 $BA^{-1}H$. $\endgroup$ May 17 '19 at 20:42

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