A Posteriori Modelling-Discretization Error Estimate for Elliptic Problems with L ∞-Coefficients

Abstract

We consider elliptic problems with complicated, discontinuous diffusion tensor A0. One of the standard approaches to numerically treat such problems is to simplify the coefficient by some approximation, say Aε, and to use standard finite elements. In [19] a combined modelling-discretization strategy has been proposed which estimates the discretization and modelling errors by a posteriori estimates of functional type. This strategy allows to balance these two errors in a problem adapted way. However, the estimate of the modelling error is derived under the assumption that the difference A0 − Aε is bounded in the L∞-norm, which requires that the approximation of the coefficient matches the discontinuities of the original coefficient. Therefore this theory is not appropriate for applications with discontinuous coefficients along complicated, curved interfaces. Based on bounds for A0 − Aε in an L q -norm with q < ∞ we generalize the combined modelling-discretization strategy to a larger class of coefficients.