A convergent adaptive algorithm for Poisson's equation
SIAM Journal on Numerical Analysis
Edge Residuals Dominate A Posteriori Error Estimates for Low Order Finite Element Methods
SIAM Journal on Numerical Analysis
Data Oscillation and Convergence of Adaptive FEM
SIAM Journal on Numerical Analysis
Local problems on stars: a posteriori error estimators, convergence, and performance
Mathematics of Computation
Adaptive Finite Element Methods with convergence rates
Numerische Mathematik
SIAM Journal on Scientific Computing
Optimality of a Standard Adaptive Finite Element Method
Foundations of Computational Mathematics
SIAM Journal on Scientific Computing
Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method
SIAM Journal on Numerical Analysis
Energy norm based a posteriori error estimation for boundary element methods in two dimensions
Applied Numerical Mathematics
Quasi-Optimal Convergence Rate of an Adaptive Discontinuous Galerkin Method
SIAM Journal on Numerical Analysis
Decay rates of adaptive finite elements with Dörfler marking
Numerische Mathematik
Convergence of adaptive BEM for some mixed boundary value problem
Applied Numerical Mathematics
Estimator reduction and convergence of adaptive BEM
Applied Numerical Mathematics
Computers & Mathematics with Applications
Hi-index | 7.29 |
We consider the solution of a second order elliptic PDE with inhomogeneous Dirichlet data by means of adaptive lowest-order FEM. As is usually done in practice, the given Dirichlet data are discretized by nodal interpolation. As model example serves the Poisson equation with mixed Dirichlet-Neumann boundary conditions. For error estimation, we use an edge-based residual error estimator which replaces the volume residual contributions by edge oscillations. For 2D, we prove convergence of the adaptive algorithm even with optimal convergence rate. For 2D and 3D, we show convergence if the nodal interpolation operator is replaced by the L^2-projection or the Scott-Zhang quasi-interpolation operator. As a byproduct of the proof, we show that the Scott-Zhang operator converges pointwise to a limiting operator as the mesh is locally refined. This property might be of independent interest besides the current application. Finally, numerical experiments conclude the work.