Calculation of a boundary law constant

Here the representative cell Z is an semi-infinite domain. The cell solution u solves the Laplace problem on Z with Dirichlet boundary condition on the bottom, periodic boundary conditions on the lateral faces, and it stabilizes exponentially fast in the upper direction. The Robin constant C we are interested in is precisely the constant to which u tabilizes.

We use Q4-finite elements on adaptively refined meshes. Additionally, also the domain extends during the adaptive calculation. Blending is used to approximate the curved boundary exactly. The adaptive scheme is controlled by a duality-based error estimator and refinement indicator.

The linear solver was a two-grid cycle, the coarse grid being Q1 finite elements on the same mesh. Smoothing was done with a vertex-centered subspace correction which can be shown to be robust. The coarse grid problem was solved with an AMG of Ruge-Stüben type.

The solution and the sequence of adaptively refined meshes can be seen in the following picture. Note that the domain has been extended in the last step.

The approximation of the Robin constant Cbl is shown in the following table: