[Zurück]

M. Feischl, M. Page, D. Praetorius:
"Convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data (GAMM 2011)";
Hauptvortrag: GAMM Jahrestagung 2011, Graz (eingeladen); 18.04.2011 - 21.04.2011; in: "PAMM: Proceedings in Applied Mathematics and Mechanics", PAMM, 11 (2011), ISSN: 1617-7061; S. 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