Fernando D. Gaspoz

Fernando D. Gaspoz, Pedro Morin

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.

Fernando D. Gaspoz, Pedro Morin, Andreas Veeser

We consider Poisson's equation with a finite number of weighted Dirac masses as a source term, together with its discretization by means of conforming finite elements. For the error in fractional Sobolev spaces, we propose residual-type a posteriori estimators with a specifically tailored oscillation and show that, on two-dimensional polygonal domains, they are reliable and locally efficient. In numerical tests, their use in an adaptive algorithm leads to optimal error decay rates.