FFT-accelerated fourth order Cartesian grid solver for elliptic problems with sharp-edged interfaces

Huanfeng Yang

doi:10.1016/j.jcp.2026.115198

FFT-accelerated fourth order Cartesian grid solver for elliptic problems with sharp-edged interfaces

2026-07-01

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DOI：10.1016/j.jcp.2026.115198

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In this paper, a fast fourth order Cartesian grid finite difference method built upon the Augmented Matched Interface and Boundary (AMIB) framework is proposed for solving two-and three-dimensional elliptic interface problems, under the assumptions that the interface is Lipschitz continuous involving sharp-edged corners and the solution is piecewise C6-continuous. For such problems, the existing Matched Interface and Boundary (MIB) methods are limited to second order accuracy, and fourth order AMIB methods that treat derivative jumps as augmented variables become ineffective in the presence of sharp-edged interfaces. This is because the fictitious values needed in the fourth order MIB and AMIB schemes cannot be generated in advance at corner points. To overcome this difficulty, the present work introduces three significant improvements. First, an improved fourth order accurate MIB (iMIB) scheme is proposed, which pairs two interface points on two edges of the corner so that fourth order MIB treatments at two interface points can be formulated simultaneously in a primary Cartesian direction, without resorting to the MIB treatments in the directions orthogonal to the primary direction. Second, the iMIB method is further augmented to construct a fourth order iAMIB method based on domain decomposition techniques. The present method retains the key advantages of the existing AMIB methods, such as the use of simple Cartesian grids, allowing FFT acceleration and handling different boundary conditions. Third, an adaptive AMIB elliptic solver is developed to attain fourth order accuracy whenever possible, while adaptively reducing the order to three when treating more complicated geometrical singularities. Extensive numerical experiments in both two and three dimensions indicate that the proposed AMIB method not only achieves fourth order accuracy for both the solution and its gradient, but also maintains overall efficiency as O(n2 log n) and O(n3 log n) on n & times; n and n & times; n & times; n uniform grids, respectively.

Keywords:

Elliptic interface problems

Sharp-edged interfaces

Domain decomposition

High-order central differences

Matched interface and boundary (MIB) method

Fast Fourier transform (FFT)

FFT-accelerated fourth order Cartesian grid solver for elliptic problems with sharp-edged interfaces

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