Convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data

Abstract

Recently, there has been a major breakthrough in the thorough mathematical understanding of convergence and quasi-optimality of h-adaptive finite element methods (AFEM) for second-order elliptic PDEs. However, the focus of the numerical analysis usually lied on model problems with homogeneous Dirichlet conditions, i.e.

-\Delta u = f in \Omega with u=0 on \Gamma=\partial\Omega,

see [Cascon et al., SINUM 2008] and the references therein. Whereas the inclusion of inhomogeneous Neumann conditions into the numerical analysis seems to be obvious, incorporating inhomogeneous Dirichlet conditions is technically more demanding.

In our talk, we consider the lowest-order AFEM for the 2D Poisson problem with mixed Dirichlet-Neumann boundary conditions. As is usually done in practice, the given Dirichlet data are discretized by nodal interpolation. We prove that this leads to a convergent adaptive scheme which recovers the best possible convergence rate with respect to the natural approximation class. Numerical experiments and some remarks on the 3D case conclude the talk.