A High Order Compact Finite Difference Scheme for Elliptic Interface Problems with Discontinuous and High-Contrast Coefficients

Abstract

The elliptic interface problems with discontinuous and high-contrast coefficients appear in many applications and often lead to huge condition numbers of the corresponding linear systems. Thus, it is highly desired to construct high order schemes to solve the elliptic interface problems with discontinuous and high-contrast coefficients. Let Γ be a smooth curve inside a rectangular region Ω. In this paper, we consider the elliptic interface problem −∇ • (a∇u) = f in Ω \ Γ with Dirichlet boundary conditions, where the coefficient a and the source term f are smooth in Ω \ Γ and the two nonzero jump condition functions [u] and [a∇u • n] across Γ are smooth along Γ. To solve such elliptic interface problems, we propose a high order compact finite difference scheme for numerically computing both the solution u and the gradient ∇u on uniform Cartesian grids without changing coordinates into local coordinates. Our numerical experiments confirm the fourth order accuracy for computing the solution u, the gradient ∇u and the velocity a∇u of the proposed compact finite difference scheme on uniform meshes for the elliptic interface problems with discontinuous and high-contrast coefficients.

Sixth Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect

“…We follow the same setup as in [14,15]. As stated in the introduction, let Ω = (l 1 , l 2 ) × (l 3 , l 4 ).…”

“…Our proposed compact finite difference scheme attains the maximum overall accuracy order everywhere on the domain with the shortest stencil support for the problem (1.1)- (1.3). Similar to [14,15], our approach is based on a critical observation regarding the inter-dependence of high order derivatives of the underlying solution. When constructing a discretization stencil, we start with a general expression that allows us to recover all possible sixth order finite difference schemes.…”

“…y) is defined in(2.10). As in[14,15], we assume that we have a parametric equation for Γ I near the base point (x * i , y * j ). I.e.,…”

Hybrid Finite Difference Schemes for Elliptic Interface Problems with Discontinuous and High-Contrast Variable Coefficients

“…In [7] we developed a compact 9-point finite difference scheme for elliptic problems, that is formally fourth order accurate away from the interface of singularity of the solution (regular points), and third order accurate in the vicinity of this interface (irregular points). The numerical experiments in [7] demonstrate that the proposed scheme is fourth order accuracy in the l 2 norm.…”

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For elliptic interface problems with discontinuous coefficients, the maximum accuracy order for compact 9-point finite difference scheme in irregular points is three [7]. The discontinuous coefficients usually have abrupt jumps across the interface curve in the porous medium of realistic problems, causing the pollution effect of numerical methods.

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