A Continuous Finite Element Method with Homotopy Vanishing Viscosity for Solving the Static Eikonal Equation
Year: 2022
Author: Yong Yang, Wenrui Hao, Yong-Tao Zhang
Communications in Computational Physics, Vol. 31 (2022), Iss. 5 : pp. 1402–1433
Abstract
We develop a second-order continuous finite element method for solving the static Eikonal equation. It is based on the vanishing viscosity approach with a homotopy method for solving the discretized nonlinear system. More specifically, the homotopy method is utilized to decrease the viscosity coefficient gradually, while Newton’s method is applied to compute the solution for each viscosity coefficient. Newton’s method alone converges for just big enough viscosity coefficients on very coarse grids and for simple 1D examples, but the proposed method is much more robust and guarantees the convergence of the nonlinear solver for all viscosity coefficients and for all examples over all grids. Numerical experiments from 1D to 3D are presented to confirm the second-order convergence and the effectiveness of the proposed method on both structured or unstructured meshes.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2021-0164
Communications in Computational Physics, Vol. 31 (2022), Iss. 5 : pp. 1402–1433
Published online: 2022-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 32
Keywords: Eikonal equation finite element method homotopy method.