Un estimador de error residual semiexplícito en cantidades de interés para un problema mecánico lineal

Abstract

We aim at defining a semi-explicit approach to estimate the error in quantities of interest associated with the Finite Element solution of a linear elasticity problem. The advocated procedure is split in two parts, an implicit error estimate for the adjoint problem and an explicit estimate assessing the error in the direct (primal) problem. The implicit part of the estimate (on the adjoint problem) embraces two phases, each consisting in projecting the error on “bubble” functional spaces. The projections are low-cost operations due to their local nature. The two phases account the error in the interior of the elements and the contribution of the elementary edges (associated with the tractions jump). The implicit character of this part is provided by the solution of the local problems in low-dimensional functional spaces. We also analyze the particular case of selecting one-dimensional functional spaces (setting the shape of the bubble and computing a scalar coefficient), which, in practice, make this part of the process explicit. The explicit part of the estimate consists in injecting in the weak primal residual the approximation of the adjoint error obtained in the first phase. This approach is similar to the DWR method but using a weak formulation of the residual and injecting an implicitly estimated adjoint error rather than a post-processed solution. The proposed methodology is validated in a numerical example