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This paper presents an improved kernel-free boundary integral method for solving elliptic partial differential equations with irregular boundaries and interfaces that are defined implicitly. The authors introduce a correction function approach that simplifies the computation of boundary and volume integrals by reducing them to simpler interface problems that can be solved efficiently using fast Fourier transforms and multigrid methods. The method demonstrates accurate results across challenging test cases including problems with high-contrast coefficients, closely spaced interfaces, and heterogeneous materials.
Why it matters
This computational technique could improve the efficiency and accuracy of simulations in engineering and physics applications involving complex geometries, such as fluid flow around irregular objects, heat transfer in composite materials, or electromagnetic problems with intricate boundary conditions. The method's ability to handle difficult cases like closely spaced interfaces makes it particularly valuable for modeling real-world scenarios with complex material distributions.
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⚠️ Preprint – Noch nicht peer-reviewed
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Abstract: This work addresses a novel version of the kernel-free boundary integral (KFBI) method for solving elliptic PDEs with implicitly defined irregular boundaries and interfaces. We focus on boundary value problems and interface problems, which are reformulated into boundary integral equations and solved with the matrix-free GMRES method. In the KFBI method, evaluating boundary and volume integrals only requires solving equivalent but much simpler interface problems in a bounding box, for which fast solvers such as FFTs and geometric multigrid methods are applicable. For the simple interface problem, a correction function is introduced for both the evaluation of right-hand side correction terms and the interpolation of a non-smooth potential function. A mesh-free collocation method is proposed to compute the correction function near the interface. The new method avoids complicated derivation for derivative jumps of the solution and is easy to implement, especially for the fourth-order method in three space dimensions. Various numerical examples are presented, including challenging cases such as high-contrast coefficients, arbitrarily close interfaces and heterogeneous interface problems. The reported numerical results verify that the proposed method is both accurate and efficient.