Non-conforming finite element methods for transmission eigenvalue problem

Abstract

:The transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering theory. In this paper, for the Helmholtz transmission eigenvalue problem, we give a weak formulation which is a nonselfadjoint linear eigenvalue problem. Based on the weak formulation, we first discuss the nonconforming finite element approximation, and prove the error estimates of the discrete eigenvalues obtained by the Adini element, Morley-Zienkiewicz element, modifiedZienkiewicz element et. al. And we report some numerical examples to validate the efficiency of our approach for solving transmission eigenvalue problem.

A spectral projection method for transmission eigenvalues

“…[23] developed a finite element method based on writing the TE as a quadratic eigenvalue problem. Other methods [10,16,19,10,30] have been proposed recently.…”

Nonconforming elements of class $L^2$ for Helmholtz transmission eigenvalue problems

“…Due to its important physical background, exploring its efficient numerical methods is a hot topic in computational mathematics. Since Colton et al [10] proposed three numerical methods, the related scholars sequentially developed many numerical methods, including conforming element methods [22,7,25,16,14], nonconforming element methods [27], spectral methods [1], C 0 IPG methods [12] and boundary element methods [17,28]. Regarding the study of conforming elements, early in 2011 Sun [22] proposed the iterative methods using conforming elements to solve the problem.…”

“…To overcome this defect, the mixed methods in [15,26] decrease the order of the problem by introducing an auxiliary variable. The nonconforming element methods in [27] make use of the weak continuity and C 0 continuity of the finite element spaces, and the C 0 IPG methods [12] introduce the inner penalty terms to keep the ellipticity of the bilinear form associated with the problem. Note that 3196 JIAYU HAN the nonconforming element methods in [27] do not cover the highly nonconforming elements of class L 2 such as the Morley element, cubic tetrahedron element and incomplete cubic tetrahedron element [23].…”

“…The nonconforming element methods in [27] make use of the weak continuity and C 0 continuity of the finite element spaces, and the C 0 IPG methods [12] introduce the inner penalty terms to keep the ellipticity of the bilinear form associated with the problem. Note that 3196 JIAYU HAN the nonconforming element methods in [27] do not cover the highly nonconforming elements of class L 2 such as the Morley element, cubic tetrahedron element and incomplete cubic tetrahedron element [23]. To fill the gap of the nonconforming element methods in [27], this paper continues to develop the new nonconforming element methods so that the corresponding finite element spaces could be discontinuous.…”

“…Note that 3196 JIAYU HAN the nonconforming element methods in [27] do not cover the highly nonconforming elements of class L 2 such as the Morley element, cubic tetrahedron element and incomplete cubic tetrahedron element [23]. To fill the gap of the nonconforming element methods in [27], this paper continues to develop the new nonconforming element methods so that the corresponding finite element spaces could be discontinuous. The new nonconforming element methods of class L 2 in this paper adopt the mixed linear formulation in [7] to discretize the eigenvalue problem.…”