Convergence rates for adaptive finite elements
Provides theoretical justification for convergence rates of adaptive finite element methods, a fundamental tool in computational PDEs.
Proved that adaptive finite element methods using newest-vertex bisection achieve quasi-optimal meshes and convergence rates for functions with point singularities, enabling optimal error equidistribution in H^1-norm.
In this article we prove that it is possible to construct, using newest-vertex bisection, meshes that equidistribute the error in $H^1$-norm, whenever the function to approximate can be decomposed as a sum of a regular part plus a singular part with singularities around a finite number of points. This decomposition is usual in regularity results of Partial Differential Equations (PDE). As a consequence, the meshes turn out to be quasi-optimal, and convergence rates for adaptive finite element methods (AFEM) using Lagrange finite elements of any polynomial degree are obtained.