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

Abstract

AbstractWe 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_{\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 difference {A_{0}-A_{\varepsilon}} becomes small with respect to the {L^{\infty}}-norm. This implies in particular that interfaces/discontinuities separating the smooth parts of {A_{0}} have to be matched exactly by the coefficient {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 difference {A_{0}-A_{\varepsilon}} is measured in the {L^{q}}-norm for some appropriate {q\in{]2,\infty[}} and, hence, the geometric resolution condition is significantly relaxed.