A high accuracy nonconforming finite element scheme for Helmholtz transmission eigenvalue problem
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In this paper, we consider a cubic H^2 nonconforming finite element scheme B_h0^3 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_h0^3 space which makes it robust for numerical discretization. For fourth order eigenvalue problem, the B_h0^3 scheme can provide O(h^2) approximation for the eigenspace in energy norm and O(h^4) approximation for the eigenvalues. We test the B_h0^3 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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