Convergence of a cartesian method for elliptic problems with immersed interfaces

Lisl Weynans (1, 2)

1 IMB - Institut de Mathématiques de Bordeaux
2 MEMPHIS - Modeling Enablers for Multi-PHysics and InteractionS

We study in this paper the convergence of a cartesian method for elliptic problems with im- mersed interfaces that was introduced in a previous paper [8]. This method is based on additional unknowns located on the interface, that are used to discretize separately the elliptic operator in each subdomain and to express the jump conditions across the interface. It was shown numerically to converge with second-order accuracy 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. This first-order version is based on the same ideas as the original method, but the discretization of the normal derivatives across the interface has only a first-order truncation error, instead of a second-order for the original method. The proof of convergence is in both cases inspired from the paper of Ciarlet [7] and takes advantage of a discrete maximum principle to obtain estimates on the coefficients of the inverse matrix.

Mots clés: discrete Green’s function; convergence; Finite-differences; cartesian method; elliptic problem; interface discontinuity; interface un- knowns

Domaines: Analyse numérique [math.NA]

Lisl Weynans. Convergence of a cartesian method for elliptic problems with immersed interfaces. [Research Report] RR-8872, INRIA Bordeaux; Univ. Bordeaux. 2016, pp.20. ⟨hal-01280283v1⟩