AFEM for elliptic problems with point sources in weighted Sobolev spaces

EDUARDO M. GARAU

Milan

Seminario; Seminarios del MOX; 2015

MOX - Department of Mathematics - Politecnico di Milano

Elliptic problems with Dirac measure source terms arise in modeling different applications as, for instance, the electric field generated by a point charge or pollutant transport and degradation in an aquatic media where, due to the different scales involved, the pollution source is modeled as supported on a single point. In spite of the fact that the solution of one such problem typically does not belong to H^1, it can be numerically approximated by standard finite elements, but there is no obvious choice for the norm to measure the error. The singular character of the solution suggests that meshes adequately refined around the delta support should be used to improve the quality of the approximation. We introduce residual type a posteriori error estimators for this kind of problems over two- and three-dimensional domains and prove a global upper bound and a local lower bound for the error measured in a weighted Sobolev space. The weight considered is a suitable positive power of the distance to the support of the Dirac delta source term. The proposed estimators are used to guide an adaptive procedure to compute the solution. Numerical experiments yield optimal meshes and very good effectivity indices.