<p>A fourth order accurate domain-decomposed Augmented Matched Interface and Boundary (AMIB) method is proposed to solve two- and three-dimensional parabolic interface problems with discontinuous diffusion coefficients. For such problems, the previously developed AMIB scheme has to utilize the geometric multigrid for inverting discrete Laplacian with non-constant diagonal coefficients. In this work, based on the domain decomposition, the interface problem is reformulated into several subproblems with constant diagonals, so that the fast Fourier transform (FFT) can be applied as the fast Poisson solver in each step of the Crank-Nicolson time stepping. Compared to other methods in this field, the proposed method offers several novel features. First, the multigrid method is an iterative algebraic solver where the number of iterations could be affected by spatial discretization, boundary conditions, and jump coefficients, while the FFT is a direct Poisson solver so that the efficiency of the proposed AMIB is significantly enhanced. Second, the interface conditions are enforced along Cartesian directions in the proposed scheme, instead of along the normal direction in the multigrid-AMIB method. An optimized corner treatment strategy is designed to simplify the fictitious value generation based on Cartesian grids. Consequently, the domain-decomposed AMIB shows a higher accuracy. Finally, by converting boundaries into immersed interfaces, both multigrid-AMIB and FFT-AMIB can handle the Dirichlet, Neumann, Robin, and mixed boundary conditions. Nevertheless, the FFT-AMIB boundary treatments are simpler and avoid using one-sided finite differences. Numerical examples indicate that the domain-decomposed AMIB achieves fourth order spatial accuracy with an overall complexity of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3039_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {O}(n^3\log {n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mn>3</mn> </msup> <mo>log</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> on a <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3039_Article_IEq2.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$n \times n \times n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> uniform grid.</p>

An Augmented Fourth Order Domain-Decomposed Method with FFT-Poisson Solver for Parabolic Interface Problems

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发表日期：

2025年9月25日

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Huanfeng Yang,
Guangqing Long,
Yiming Ren,
Shan Zhao

摘要

A fourth order accurate domain-decomposed Augmented Matched Interface and Boundary (AMIB) method is proposed to solve two- and three-dimensional parabolic interface problems with discontinuous diffusion coefficients. For such problems, the previously developed AMIB scheme has to utilize the geometric multigrid for inverting discrete Laplacian with non-constant diagonal coefficients. In this work, based on the domain decomposition, the interface problem is reformulated into several subproblems with constant diagonals, so that the fast Fourier transform (FFT) can be applied as the fast Poisson solver in each step of the Crank-Nicolson time stepping. Compared to other methods in this field, the proposed method offers several novel features. First, the multigrid method is an iterative algebraic solver where the number of iterations could be affected by spatial discretization, boundary conditions, and jump coefficients, while the FFT is a direct Poisson solver so that the efficiency of the proposed AMIB is significantly enhanced. Second, the interface conditions are enforced along Cartesian directions in the proposed scheme, instead of along the normal direction in the multigrid-AMIB method. An optimized corner treatment strategy is designed to simplify the fictitious value generation based on Cartesian grids. Consequently, the domain-decomposed AMIB shows a higher accuracy. Finally, by converting boundaries into immersed interfaces, both multigrid-AMIB and FFT-AMIB can handle the Dirichlet, Neumann, Robin, and mixed boundary conditions. Nevertheless, the FFT-AMIB boundary treatments are simpler and avoid using one-sided finite differences. Numerical examples indicate that the domain-decomposed AMIB achieves fourth order spatial accuracy with an overall complexity of $\mathcal {O}(n^3\log {n})$ $O (n^{3} log n)$ on a $n \times n \times n$ $n \times n \times n$ uniform grid.