Abstract
For solving the Helmholtz transmission eigenvalue problem, we use the mixed formulation of Cakoni et al. to construct a new nonconforming element discretization. Based on the discretization, this paper first discuss the nonconforming element methods of class $ L^2 $, and prove the error estimates of the discrete eigenvalues obtained by the cubic tetrahedron element, incomplete cubic tetrahedral element and Morley element et al. We report some numerical examples using the nonconforming elements mixed with linear Lagrange element to show that our discretization can obtain the transmission eigenvalues of higher accuracy in 3D domains than the nonconforming element discretization in the existing literature.
Highlights
The transmission eigenvalue problem arising in the inverse scattering theory is an important problem in acoustic scattering [6, 11, 21]
With a weak formulation different from (2.7)-(2.8), we have studied some nonconforming finite element approximations
In view of the discontinuity of nonconforming element considered, we have to give the linear boundedness of the piecewise form of B(·, ·)
Summary
The transmission eigenvalue problem arising in the inverse scattering theory is an important problem in acoustic scattering [6, 11, 21]. The new nonconforming element methods of class L2 in this paper adopt the mixed linear formulation in [7] to discretize the eigenvalue problem. Due to the discontinuity of nonconforming element spaces, we have to apply the spectral approximation theory of [2] in the union of the piecewise smooth Sobolev spaces, which needs to be completed. This increases the difficulty of our theory analysis, we overcome it and explore a new analysis approach of applying the spectral approximation theory in piecewise smooth Sobolev spaces Another obvious feature of our error analysis is the application of the projector-mean operator in [20], which plays a crucial role in the proof of the error of consistence terms and eigenvalues.
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