A fourth-order Cartesian grid method for elliptic problems with sharp-edged interfaces

Yiming Ren

doi:10.1016/j.jcp.2026.114746

2026-02-04

PRE

DOI：10.1016/j.jcp.2026.114746

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In this paper, we propose the first fourth-order Cartesian grid method built upon the augmented matched interface and boundary (AMIB) framework for solving two-dimensional elliptic interface problems with Lipschitz continuous interfaces involving sharp-edged corners. By using non-body-fitted meshes, traditional Cartesian grid methods are limited to be second-order accurate near sharp-edged corners, because there are insufficient grid nodes to support high-order approximations. To improve the efficiency and stability of interface methods, an augmented formulation is commonly used in the literature, which introduces auxiliary variables, such as Cartesian derivative jumps, so that the discrete Laplacian could be inverted by fast Poisson solvers, including fast Fourier transform and multigrid. In this work, a new correction to the fourth-order central difference approximation of the Laplacian is proposed, by treating the fictitious jumps, which measure the differences between fictitious values (FVs) and function values at irregular grid points near the interface, as the auxiliary variables in the AMIB method. Besides being simpler and involving less approximations, this new correction allows us to generate FVs in an entirely different manner, i.e., one FV could potentially depend on all other unknown FVs. This makes a lot of FVs available near sharp-edged corners to discretize the jump conditions, so that fourth-order approximations could be accomplished for problems with piecewise C6 continuous solutions. These unknown fictitious jumps will be solved together with function values in an enlarged linear system by using a Schur complement procedure combined with fast Poisson solvers. Numerical experiments demonstrate that the AMIB-FV scheme achieves fourth-order accuracy for both solutions and their gradients, while maintaining the overall efficiency as O(n2log n) or O(n2) for an n × n uniform grid, in solving complex interface problems with mixed boundary conditions. Moreover, the condition numbers of the AMIB-FV scheme are significantly smaller than those of the existing AMIB methods.