The smooth forcing extension method: A high-order technique for solving elliptic equations on complex domains

JOURNAL OF COMPUTATIONAL PHYSICS

卷:439

Article

Qadeer, Saad¹  Griffith, Boyce E.^1,2,3 

[1] Univ N Carolina, Dept Math, Chapel Hill, NC 27515 USA

[2] Univ N Carolina, Dept Appl Phys Sci, Chapel Hill, NC 27515 USA

[3] Univ N Carolina, Dept Biomed Engn, Chapel Hill, NC 27515 USA

关键词: Elliptic equations;    Fourier continuation;    Fixed Cartesian grid methods;    Immersed boundary method;    Immersed boundary smooth extension;    Non-uniform fast Fourier transform;   

DOI  :  10.1016/j.jcp.2021.110390

来源: Elsevier

【 摘 要 】

High-order numerical methods for solving elliptic equations over arbitrary domains typically require specialized machinery, such as high-quality conforming grids for finite elements method, and quadrature rules for boundary integral methods. These tools make it difficult to apply these techniques to higher dimensions. In contrast, fixed Cartesian grid methods, such as the immersed boundary (IB) method, are easy to apply and generalize, but typically are low-order accurate. In this study, we introduce the Smooth Forcing Extension (SFE) method, a fixed Cartesian grid technique that builds on the insights of the IB method, and allows one to obtain arbitrary orders of accuracy. Our approach relies on a novel Fourier continuation method to compute extensions of the inhomogeneous terms to any desired regularity. This is combined with the highly accurate Non-Uniform Fast Fourier Transform for interpolation operations to yield a fast and robust method. Numerical tests confirm that the technique performs precisely as expected on one-dimensional test problems. In higher dimensions, the performance is even better, in some cases yielding sub-geometric convergence. We also demonstrate how this technique can be applied to solving parabolic problems and for computing the eigenvalues of elliptic operators on general domains, in the process illustrating its stability and amenability to generalization. Published by Elsevier Inc.

【 授权许可】

Free