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

Abstract

AbstractWe consider elliptic problems with complicated, discontinuous diffusion tensorA0{A_{0}}. One of the standard approaches to numerically treat such problems is to simplify the coefficient by some approximation, sayAε{A_{\varepsilon}}, 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 was derived under the assumption that the differenceA0-Aε{A_{0}-A_{\varepsilon}}becomes small with respect to theL∞{L^{\infty}}-norm. This implies in particular that interfaces/discontinuities separating the smooth parts ofA0{A_{0}}have to be matched exactly by the coefficientAε{A_{\varepsilon}}. Therefore the efficient application of that theory to problems with complicated or curved interfaces is limited. In this paper, we will present a refined theory, where the differenceA0-Aε{A_{0}-A_{\varepsilon}}is measured in theLq{L^{q}}-norm for some appropriateq∈]2,∞[{q\in{]2,\infty[}}and, hence, the geometric resolution condition is significantly relaxed.