Abstract
In this paper, a type of full multigrid method is proposed to solve non-selfadjoint Steklov eigenvalue problems. Multigrid iterations for corresponding selfadjoint and positive definite boundary value problems generate proper iterate solutions that are subsequently added to the coarsest finite element space in order to improve approximate eigenpairs on the current mesh. Based on this full multigrid, we propose a new type of adaptive finite element method for non-selfadjoint Steklov eigenvalue problems. We prove that the computational work of these new schemes are almost optimal, the same as solving the corresponding positive definite selfadjoint boundary value problems. In this case, these type of iteration schemes certainly improve the overfull efficiency of solving the non-selfadjoint Steklov eigenvalue problem. Some numerical examples are provided to validate the theoretical results and the efficiency of this proposed scheme.
Highlights
Inverse scattering problems for inhomogeneous media have many applications, such as medical imaging and nondestructive testing and so on
Compared with the transmission eigenvalue problem, non-selfadjoint Steklov eigenvalues associated with the scattering problem have many advantages [30], and the potential to work for a wider class of problems, such as the surface waves, mechanical oscillators immersed in a viscous fluid and the vibration modes of a structure in contact with an incompressible fluid [10, 11, 27, 31]
The aim of this paper is to present a full multigrid method [15, 26, 44] for solving non-selfadjoint Steklov eigenvalue problems based on the combination of the multilevel correction method and the multigrid iteration for boundary value problems
Summary
Inverse scattering problems for inhomogeneous media have many applications, such as medical imaging and nondestructive testing and so on. The aim of this paper is to present a full multigrid method [15, 26, 44] (sometimes referred to as nested finite element method) for solving non-selfadjoint Steklov eigenvalue problems based on the combination of the multilevel correction method and the multigrid iteration for boundary value problems. Some multigrid iteration steps are used to get an approximate solution In this new version of multigrid method, solving non-selfadjoint Steklov eigenvalue problems will not be much more difficult than the multigrid scheme for the corresponding positive definite selfadjoint boundary value problems. We give a new type of AFEM based on full multigrid In this method, solving non-selfadjoint Steklov eigenvalue problem only includes solving the associated positive definite selfadjoint boundary value problems on a series of adaptively refined partitions by multigrid method and the non-selfadjoint Steklov eigenvalue problem with a coarse mesh. Some concluding remarks are given in the last section
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