A high accuracy nonconforming finite element scheme for Helmholtz transmission eigenvalue problem

2019/10/02 by Yingxia Xi, Xi, Yingxia, Xia Ji +3
Engineering · #31A30 #65N30 #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.1910.00898

openalex publication_date 2019/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider a cubic H² nonconforming finite element scheme B_h0³ which does not correspond to a locally defined finite element with Ciarlet's triple but admit a set of local basis functions. For the first time, we deduce and write out the expression of basis functions explicitly. Distinguished from the most nonconforming finite element methods, (δΔ_h⋅,Δ_h⋅) with non-constant coefficient δ>0 is coercive on the nonconforming B_h0³ space which makes it robust for numerical discretization. For fourth order eigenvalue problem, the B_h0³ scheme can provide O(h²) approximation for the eigenspace in energy norm and O(h⁴) approximation for the eigenvalues. We test the B_h0³ scheme on the vary-coefficient bi-Laplace source and eigenvalue problem, further, transmission eigenvalue problem. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed scheme.

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