A high order compact finite difference scheme for elliptic interface problems with discontinuous and high-contrast coefficients

Feng, Qiwei; Han, Bin; Minev, Peter

doi:10.1016/j.amc.2022.127314

Cited by 7 publications

(6 citation statements)

References 33 publications

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“…u (1,0) u (1,1) u (1,2) u (1,3) u (1,4) u (1,5) u (1,6) u (2,0) u (2,1) u (2,2) u (2,3) u (2,4) u (2,5) u (3,0) u (3,1) u (3,2) u (3,3) u (3,4) u (4,0) u (4,1) u (4,2) u (4,3) u (5,0) u (5,1) u (5,2) u (6,0) u (6,1) u (7,0) u (0,0) u (0,1) u (0,2) u (0,3) u (0,4) u (0,5) u (0,6) u (0,7)…”

Section: Hybrid Fdms On Uniform Cartesian Grids For the Elliptic Inte...unclassified

“…u (1,0) u (1,1) u (1,2) u (1,3) u (1,4) u (1,5) u (1,6) Figure 2. The illustration for (2.6)-(2.12) with M = 6.…”

Section: Hybrid Fdms On Uniform Cartesian Grids For the Elliptic Inte...mentioning

confidence: 99%

“…For the anisotropic elliptic interface problems with discontinuous and matrix coefficients, [2] proposed a new second-order finite element-finite difference (FE-FD) method. [6] developed a 9-point compact FDM for elliptic interface problems with discontinuous scalar coefficients, that is formally fourth-order consistent away from the interface of singularity of the solution (regular points), and third-order consistent in the vicinity of the interface (irregular points). In the present paper we derive a 9-point compact scheme that has the sixth-order consistency at regular points.…”

Section: Introduction and Motivationsmentioning

confidence: 99%

“…Furthermore, we also propose a recursive solver to efficiently derive the stencil coefficients of the proposed scheme. The resulting sixth-order hybrid scheme shows a significantly improved numerical performance with a slight increase in its complexity over the fourth-order scheme in [6]. Theoretically, our hybrid FDMs use high order (partial) derivatives of the coefficient function a, the source term f , the interface curve Γ, the two jump functions g, g Γ on Γ, and boundary data.…”

Section: Introduction and Motivationsmentioning

confidence: 99%

See 3 more Smart Citations

Compact 9-point finite difference methods with high accuracy order and/or M-matrix property for elliptic cross-interface problems

Feng

Han

Minev

2023

Journal of Computational and Applied Mathematics

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Section: Hybrid Fdms On Uniform Cartesian Grids For the Elliptic Inte...unclassified

“…u (1,0) u (1,1) u (1,2) u (1,3) u (1,4) u (1,5) u (1,6) Figure 2. The illustration for (2.6)-(2.12) with M = 6.…”

Section: Hybrid Fdms On Uniform Cartesian Grids For the Elliptic Inte...mentioning

confidence: 99%

Section: Introduction and Motivationsmentioning

confidence: 99%

Section: Introduction and Motivationsmentioning

confidence: 99%

See 2 more Smart Citations

Compact 9-point finite difference methods with high accuracy order and/or M-matrix property for elliptic cross-interface problems

Feng

Han

Minev

2023

Journal of Computational and Applied Mathematics

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“…Finally we should mention, that even in the relatively simple case of elliptic interface problems involving a smooth non-intersecting interface of coefficient jump, the theoretical proof of convergence of the various proposed finite difference schemes is usually missing. The only exceptions are presented in [21] using discrete maximum principle for a second order scheme, and [12] using numerically verified discrete maximum principle for a fourth order scheme. The compact 9-point schemes considered in the present paper possess the M -matrix property, that guarantees the discrete maximum principle for the numerical solution.…”

Section: Introduction and Problem Formulationmentioning

confidence: 99%

Compact 9-Point Finite Difference Methods with High Accuracy Order and/or M-Matrix Property for Elliptic Cross-Interface Problems

Feng¹,

Han²,

Minev³

2022

Preprint

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In this paper we develop finite difference schemes for elliptic problems with piecewise continuous coefficients that have (possibly huge) jumps across fixed internal interfaces. In contrast with such problems involving one smooth non-intersecting interface, that have been extensively studied, there are very few papers addressing elliptic interface problems with intersecting interfaces of coefficient jumps. It is well known that if the values of the permeability in the four subregions around a point of intersection of two such internal interfaces are all different, the solution has a point singularity that significantly affects the accuracy of the approximation in the vicinity of the intersection point. In the present paper we propose a fourth-order 9-point finite difference scheme on uniform Cartesian meshes for an elliptic problem whose coefficient is piecewise constant in four rectangular subdomains of the overall two-dimensional rectangular domain. Moreover, for the special case when the intersecting point of the two lines of coefficient jumps is a grid point, such a compact scheme, involving relatively simple formulas for computation of the stencil coefficients, can even reach sixth order of accuracy. Furthermore, we show that the resulting linear system for the special case has an M -matrix, and prove the theoretical sixth order convergence rate using the discrete maximum principle. Our numerical experiments demonstrate the fourth (for the general case) and sixth (for the special case) accuracy orders of the proposed schemes. In the general case, we derive a compact third-order finite difference scheme, also yielding a linear system with an M -matrix. In addition, using the discrete maximum principle, we prove the third order convergence rate of the scheme for the general elliptic cross-interface problem.

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A FFT accelerated fourth order finite difference method for solving three-dimensional elliptic interface problems

Ren

Zhao

2023

Journal of Computational Physics

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A high order compact finite difference scheme for elliptic interface problems with discontinuous and high-contrast coefficients

Cited by 7 publications

References 33 publications

Compact 9-point finite difference methods with high accuracy order and/or M-matrix property for elliptic cross-interface problems

Compact 9-point finite difference methods with high accuracy order and/or M-matrix property for elliptic cross-interface problems

Compact 9-Point Finite Difference Methods with High Accuracy Order and/or M-Matrix Property for Elliptic Cross-Interface Problems

A FFT accelerated fourth order finite difference method for solving three-dimensional elliptic interface problems

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