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

Monika Weymuth; Stefan Sauter; Sergey Repin

doi:10.1515/cmam-2017-0015

Published by De Gruyter June 21, 2017

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

Monika Weymuth , Stefan Sauter and Sergey Repin

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2017-0015

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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∈]2,∞[ and, hence, the geometric resolution condition is significantly relaxed.

Keywords: A Posteriori Error Estimation; Elliptic Regularity; Model Simplification

MSC 2010: 65N30; 65N15; 35J25

A Interpolation Estimates

Lemma A.1.

Let 2<r<t<∞ and θ∈(0,1) be such that

1r=θ2+1-θt.

Then if u∈Lt⁢(Ω), we have the estimate

(A.1)∥u∥r,Ω≤∥u∥2,Ωθ⁢∥u∥t,Ω1-θ.

A proof can be found in [1]. Note that inequality (A.1) also holds for vector-valued functions.

Theorem A.2 (Marcinkiewicz interpolation theorem).

Let S be a linear mapping from Lq⁢(Ω)∩Lr⁢(Ω) into itself, 1≤q<r<∞ and suppose that there are constants S1 and S2 such that

μS⁢f⁢(t)≤(S1⁢∥f∥q,Ωt)q,μS⁢f⁢(t)≤(S2⁢∥f∥r,Ωt)r

for all f∈Lq⁢(Ω)∩Lr⁢(Ω) and t>0. Then S extends as a bounded linear mapping from Lp⁢(Ω) into itself for any p such that q<p<r and

∥S⁢f∥p,Ω≤2⁢(pp-q+pr-p)1p⁢S1α⁢S21-α⁢∥f∥p,Ω

for all f∈Lq⁢(Ω)∩Lr⁢(Ω), where

1p=αq+1-αr.

For a proof we refer to [12, Theorem 9.8].

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Received: 2017-4-6

Revised: 2017-5-18

Accepted: 2017-5-19

Published Online: 2017-6-21

Published in Print: 2017-7-1