M. Feischl, M. Page,D. Praetorius:

"Convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data (GAMM 2011)";

Keynote Lecture: GAMM Jahrestagung 2011, Graz (invited); 04-18-2011 - 04-21-2011; in: "PAMM: Proceedings in Applied Mathematics and Mechanics", PAMM, 11 (2011), ISSN: 1617-7061; 769 - 772.

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.

http://dx.doi.org/10.1002/pamm.201110374

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