FoCM 2014 conference

Workshop A5 - Multiresolution and Adaptivity in Numerical PDEs

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A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces

Juan Pablo Agnelli

Universidad Nacional de Córdoba and CIEM-CONICET, Argentina   -

In this work we develop a posteriori error estimates for second order linear elliptic problems with point sources in two- and three-dimensional domains. We prove a global upper bound and a local lower bound for the error measured in a weighted Sobolev space. The weight considered is a (positive) power of the distance to the support of the Dirac delta source term, and belongs to the Muckenhoupt's class $A_2$. The theory hinges on local approximation properties of either Cl\'ement or Scott-Zhang interpolation operators, without need of modifications, and makes use of weighted estimates for fractional integrals and maximal functions. Numerical experiments with an adaptive algorithm yield optimal meshes and very good effectivity indices.

Joint work with Eduardo M. Garau (Universidad Nacional del Litoral and IMAL-CONICET, Argentina) and Pedro Morin (Universidad Nacional del Litoral and IMAL-CONICET, Argentina).

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