A convergent adaptive algorithm for Poisson's equation
SIAM Journal on Numerical Analysis
Local and parallel finite element algorithms based on two-grid discretizations
Mathematics of Computation
A New Paradigm for Parallel Adaptive Meshing Algorithms
SIAM Journal on Scientific Computing
Data Oscillation and Convergence of Adaptive FEM
SIAM Journal on Numerical Analysis
Convergence of Adaptive Finite Element Methods
SIAM Review
Elliptic Reconstruction and a Posteriori Error Estimates for Parabolic Problems
SIAM Journal on Numerical Analysis
Generalized Green's Functions and the Effective Domain of Influence
SIAM Journal on Scientific Computing
Convergence of Adaptive Finite Element Methods for General Second Order Linear Elliptic PDEs
SIAM Journal on Numerical Analysis
Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method
SIAM Journal on Numerical Analysis
Hi-index | 0.00 |
A number of works concerning rigorous convergence theory for adaptive finite element methods (AFEMs) for controlling global energy errors have appeared in recent years. However, many practical situations demand AFEMs designed to efficiently compute quantities which depend on the unknown solution only on some subset of the overall computational domain. In this work we prove convergence of an AFEM for controlling local energy errors. The first step in our convergence proof is the construction of novel a posteriori error estimates for controlling a weighted local energy error. This weighted local energy notion admits versions of standard ingredients for proving convergence of AFEMs such as quasi-orthogonality and error contraction, but modulo “pollution terms” which use weaker norms to measure effects of global solution properties on the local energy error. We then prove several convergence results for AFEMs based on various marking strategies, including a contraction result in the case of convex polyhedral domains.