In this paper, we present a multi-level mixed element scheme for the Helmholtz transmission eigenvalue problem on polygonal domains that are not necessarily able to be covered by rectangle grids. We first construct an equivalent linear mixed formulation of the transmission eigenvalue problem and then discretize it with Lagrangian finite elements of low regularities. The proposed scheme admits a natural nested discretization, based on which we construct a multi-level scheme. Optimal convergence rate and optimal computational cost can be obtained with the scheme.2000 Mathematics Subject Classification. 65N25,65N30,47B07. Key words and phrases. Mixed finite element method; transmission eigenvalue; multi-level scheme.X.
Abstract. In this paper, we discuss approximating the eigenvalue problem of biharmonic equation. We first present an equivalent mixed formulation which admits amiable nested discretization.Then, we construct multi-level finite element schemes by implementing the algorithm as in [33] to the nested discretizations on series of nested grids. The multi-level mixed scheme for biharmonic eigenvalue problem possesses optimal convergence rate and optimal computational cost. Both theoretical analysis and numerical verifications are presented.
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