The Extrinsic Enriched Finite Element Method with Appropriate Enrichment Functions for the Helmholtz Equation

Yingbin Chai, Kangye Huang, Shangpan Wang, Zhichao Xiang and Guanjun Zhang ()
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Yingbin Chai: Laboratory of High Performance Ship Technology, Ministry of Education, Wuhan University of Technology, Wuhan 430063, China
Kangye Huang: School of Naval Architecture, Ocean and Energy Power Engineering, Wuhan University of Technology, Wuhan 430063, China
Shangpan Wang: School of Naval Architecture, Ocean and Energy Power Engineering, Wuhan University of Technology, Wuhan 430063, China
Zhichao Xiang: School of Naval Architecture, Ocean and Energy Power Engineering, Wuhan University of Technology, Wuhan 430063, China
Guanjun Zhang: Laboratory of High Performance Ship Technology, Ministry of Education, Wuhan University of Technology, Wuhan 430063, China

Mathematics, 2023, vol. 11, issue 7, 1-25

Abstract: The traditional finite element method (FEM) could only provide acceptable numerical solutions for the Helmholtz equation in the relatively small wave number range due to numerical dispersion errors. For the relatively large wave numbers, the corresponding FE solutions are never adequately reliable. With the aim to enhance the numerical performance of the FEM in tackling the Helmholtz equation, in this work an extrinsic enriched FEM (EFEM) is proposed to reduce the inherent numerical dispersion errors in the standard FEM solutions. In this extrinsic EFEM, the standard linear approximation space in the linear FEM is enriched extrinsically by using the polynomial and trigonometric functions. The construction of this enriched approximation space is realized based on the partition of unity concept and the highly oscillating features of the Helmholtz equation in relatively large wave numbers can be effectively captured by the employed specially-designed enrichment functions. A number of typical numerical examples are considered to examine the ability of this extrinsic EFEM to control the dispersion error for solving Helmholtz problems. From the obtained numerical results, it is found that this extrinsic EFEM behaves much better than the standard FEM in suppressing the numerical dispersion effects and could provide much more accurate numerical results. In addition, this extrinsic EFEM also possesses higher convergence rate than the conventional FEM. More importantly, the formulation of this extrinsic EFEM can be formulated quite easily without adding the extra nodes. Therefore, the present extrinsic EFEM can be regarded as a competitive alternative to the traditional finite element approach in dealing with the Helmholtz equation in relatively high frequency ranges.

Keywords: Helmholtz equation; finite element method (FEM); meshfree method; partition of unity; pollution error (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

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