A posteriori modelling-discretization error estimate for elliptic problems with L ∞-Coefficients

Weymuth, Monika; Sauter, Stefan A; Repin, Sergey (2017). A posteriori modelling-discretization error estimate for elliptic problems with L ∞-Coefficients. Computational Methods in Applied Mathematics, 17(3):515-531.

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 was derived under the assumption that the difference A0 - Aϵ becomes small with respect to the L∞-norm. This implies in particular that interfaces/discontinuities separating the smooth parts of A0 have to be matched exactly by the coefficient Aϵ. 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 A0 - Aϵ is measured in the Lq-norm for some appropriate q ϵ ]2 , ∞[ and, hence, the geometric resolution condition is significantly relaxed.

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 was derived under the assumption that the difference A0 - Aϵ becomes small with respect to the L∞-norm. This implies in particular that interfaces/discontinuities separating the smooth parts of A0 have to be matched exactly by the coefficient Aϵ. 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 A0 - Aϵ is measured in the Lq-norm for some appropriate q ϵ ]2 , ∞[ and, hence, the geometric resolution condition is significantly relaxed.

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