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Volume 31, Issue 5
A Continuous Finite Element Method with Homotopy Vanishing Viscosity for Solving the Static Eikonal Equation

Yong Yang, Wenrui Hao & Yong-Tao Zhang

Commun. Comput. Phys., 31 (2022), pp. 1402-1433.

Published online: 2022-05

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  • 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.

  • AMS Subject Headings

65N06, 65N12, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-31-1402, author = {Yang , YongHao , Wenrui and Zhang , Yong-Tao}, title = {A Continuous Finite Element Method with Homotopy Vanishing Viscosity for Solving the Static Eikonal Equation}, journal = {Communications in Computational Physics}, year = {2022}, volume = {31}, number = {5}, pages = {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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0164}, url = {http://global-sci.org/intro/article_detail/cicp/20509.html} }
TY - JOUR T1 - A Continuous Finite Element Method with Homotopy Vanishing Viscosity for Solving the Static Eikonal Equation AU - Yang , Yong AU - Hao , Wenrui AU - Zhang , Yong-Tao JO - Communications in Computational Physics VL - 5 SP - 1402 EP - 1433 PY - 2022 DA - 2022/05 SN - 31 DO - http://doi.org/10.4208/cicp.OA-2021-0164 UR - https://global-sci.org/intro/article_detail/cicp/20509.html KW - Eikonal equation, finite element method, homotopy method. AB -

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.

Yong Yang, Wenrui Hao & Yong-Tao Zhang. (2022). A Continuous Finite Element Method with Homotopy Vanishing Viscosity for Solving the Static Eikonal Equation. Communications in Computational Physics. 31 (5). 1402-1433. doi:10.4208/cicp.OA-2021-0164
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