Convergence of a cartesian method for elliptic problems with immersed interfaces

Lisl Weynans 1, 2
2 MEMPHIS - Modeling Enablers for Multi-PHysics and InteractionS
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : We study in this paper the convergence of a Cartesian method for elliptic problems with immersed interfaces that was introduced in a previous paper [7]. This method is based on additional unknowns located on the interface, used to express the jump conditions across the interface and to 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. Indeed, 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 uses discrete Green’s functions and takes advantage of a discrete maximum principle to obtain estimates on the coefficients of the inverse matrix.
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[Research Report] RR-8872, INRIA Bordeaux; Univ. Bordeaux. 2017, pp.24
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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-01280283v2〉

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