A Simple Low-Degree Optimal Finite Element Scheme for the Elastic Transmission Eigenvalue Problem

2021 ◽  
Vol 30 (4) ◽  
pp. 1061-1082
Author(s):  
global sci
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
Transmission Eigenvalue ◽  
Low Degree ◽  
Transmission Eigenvalue Problem

scholarly journals An Adaptive Finite Element Scheme for the Hellinger–Reissner Elasticity Mixed Eigenvalue Problem

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Fleurianne Bertrand ◽  
Daniele Boffi ◽  
Rui Ma
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Degrees Of Freedom ◽  
Elasticity Problem ◽  
Error Estimator ◽  
Adaptive Finite Element ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
Mixed Eigenvalue

Abstract In this paper, we study the approximation of eigenvalues arising from the mixed Hellinger–Reissner elasticity problem by using a simple finite element introduced recently by one of the authors. We prove that the method converges when a residual type error estimator is considered and that the estimator decays optimally with respect to the number of degrees of freedom. A postprocessing technique originally proposed in a different context is discussed and tested numerically.

scholarly journals A multi-level mixed element scheme of the two-dimensional Helmholtz transmission eigenvalue problem

2018 ◽  
Vol 40 (1) ◽  
pp. 686-707 ◽  
Author(s):  
Yingxia Xi ◽  
Xia Ji ◽  
Shuo Zhang
Keyword(s):  
Eigenvalue Problem ◽  
Computational Cost ◽  
Optimal Convergence Rate ◽  
Element Scheme ◽  
Polygonal Domains ◽  
Transmission Eigenvalue ◽  
Transmission Eigenvalue Problem ◽  
Rectangular Grids ◽  
Multi Level ◽  
Mixed Element

Abstract 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 rectangular 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.

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