A Posteriori Modelling-Discretization Error Estimate for Elliptic Problems with L ^\infty -Coefficients

Abstract

We consider elliptic problems with complicated, discontinuous diffusion tensor $A_0$. One of the standard approaches to numerically treat such problems is to simplify the coefficient by some approximation, say $A_{\epsilon }$, and to use standard finite elements. In \cite{Repin2012} 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 $A_0-A_{\epsilon }$ is bounded in the $L^{\infty }$-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 \textit{complicated, curved} interfaces. Based on bounds for $A_0-A_{\epsilon }$ in an $L^q$-norm with $q<\infty$ we generalize the combined modelling-discretization strategy to a larger class of coefficients.