A highly-accurate finite element method with exponentially compressed meshes for the solution of the Dirichlet problem of the generalized Helmholtz equation with corner singularities

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:361

Article

Celiker, Emine^1,2  Lin, Ping¹ 

[1] Univ Dundee, Div Math, 23 Perth Rd, Dundee DD1 4HN, Scotland

[2] Univ Lincoln, Dept Life Sci, Lincoln LN6 7TS, England

关键词: Mesh refinement;    Helmholtz equation;    Singularity problem;    Finite element method;    Error analysis;   

DOI  :  10.1016/j.cam.2019.04.004

来源: Elsevier

【 摘 要 】

In this study, a highly-accurate, conforming finite element method is developed and justified for the solution of the Dirichlet problem of the generalized Helmholtz equation on domains with re-entrant corners. The k - th order Lagrange elements are used for the discretization of the variational form of the problem on exponentially compressed polar meshes employed in the neighbourhood of the corners whose interior angle is alpha pi, alpha not equal 1/2, and on the triangular and curved mesh formed in the remainder of the polygon. The exponentially compressed polar meshes are constructed such that they are transformed to square meshes using the Log-Polar transformation, simplifying the realization of the method significantly. For the error bound between the exact and the approximate solution obtained by the proposed method, an accuracy of O(h(k)), h mesh size and k >= 1 an integer, is obtained in the H-1-norm. Numerical experiments are conducted to support the theoretical analysis made. The proposed method can be applied for dealing with the corner singularities of general nonlinear parabolic partial differential equations with semi-implicit time discretization. Crown Copyright (C) 2019 Published by Elsevier B.V. All rights reserved.

【 授权许可】

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