A High Accuracy Nonconforming Finite Element Scheme for Helmholtz Transmission Eigenvalue Problem

In this paper, we consider a cubic H² nonconforming finite element scheme Bh03 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 Bh03 space which makes it robust for numerical discretization. For fourth order eigenvalue problem, the Bh03 scheme can provide O(h²) approximation for the eigenspace in energy norm and O(h⁴) approximation for the eigenvalues. We test the Bh03 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.