Estima de error residual explícita para cantidades de interés utilizando funciones burbuja

Abstract

A new goal-oriented residual a posteriori error estimator is introduced for elliptic problems. The proposed estimate is based on using bubble functions over both elements and edges. The error representation of the quantity of interest prescribed by the user is performed using an adjoint problem. These quantities of interest are either averaged values of the solution (mean temperature) or point values as, for instance, the displacement of a given point. The estimation procedure is organized in two phases. First, the error is approximated in the interior of the elements projecting it into a given bubble function. The first contribution to the overall error (interior estimate) is just the sum of all these projections. Second, a new family of bubble functions is considered, each associated with one edge of the finite element mesh. The remaining part of the error is projected into this new family of functions. This second error contribution (edge estimate) is added to the previous one to obtain the complete estimate. The present estimator is independent of the typical unknown constants appearing in explicit residual type estimates. Despite of its explicit character, the present estimator gets rid of these constants by properly combining the residuals in the adjoint and direct problems, although both are explicitly computed.

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