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Convergence of a cartesian method for elliptic problems with immersed interfaces

Lisl Weynans 1
1 CARMEN - Modélisation et calculs pour l'électrophysiologie cardiaque
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest, IHU-LIRYC
Abstract : We study in this report the convergence of a Cartesian method for elliptic problems with immersed interfaces that was presented in a previous paper. This method is based on additional unknowns located on the interface, used to express the jump conditions across the interface and discretize the elliptic operator in each subdomain separately. It is numerically second-order accurate in L¥-norm. This paper is a step toward the convergence proof of this method. We prove the convergence of the method in two cases: the original second-order method in one dimension, and a first-order version in two dimensions. The proof of convergence takes advantage of a discrete maximum principle to obtain estimates on the coefficients of the inverse matrix. More precisely, we obtain estimates for the sums of the coefficients of several blocks of the inverse matrix. Associated to the consistency error, which has different leading orders throughout the domain, these estimates lead to the convergence results.
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Submitted on : Tuesday, July 16, 2019 - 3:32:00 PM
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Lisl Weynans. Convergence of a cartesian method for elliptic problems with immersed interfaces. [Research Report] RR-8872, INRIA Bordeaux; Univ. Bordeaux. 2017, pp.24. ⟨hal-01280283v6⟩

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