A posteriori error estimates for elliptic problems with Dirac delta source terms in weighted Sobolev spaces

JUAN PABLO AGNELLI, EDUARDO M. GARAU, PEDRO MORIN

Córdoba

Congreso; IV Congreso Latinoamericano de Matemáticos; 2012

UMA - UMALCA

Elliptic problems with Dirac delta sources arise in different applications such as %, for instance, modeling pollutant transport and degradation in an aquatic media cite{ArBeRo_advection} or multiscale modeling of blood flow through tissues cite{D´A_Q}. 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. The singular character of the solution suggests that meshes adequately refined around the delta support should be used to improve the quality of the approximation. In this work, we introduce residual type a posteriori error estimators for these kind of problems in weighted Sobolev spaces. The error is measured in $W^{1,2}_{d^{alpha}}$, where $d(x)=dist(x,x_0)$, $x_0$ is the point where the Dirac delta is located and $alpha in (0,1/2)$. This space is ``larger" than $H^{1}$ and seems to be more appropriate than the $W^{1,p}$ spaces with $p