21,425 research outputs found
A stabilized finite element method for the two-field and three-field Stokes eigenvalue problems
In this paper, the stabilized finite element approximation of the Stokes eigenvalue problems is considered for both the two-field (displacement–pressure) and the three-field (stress–displacement–pressure) formulations. The method presented is based on a subgrid scale concept, and depends on the approximation of the unresolvable scales of the continuous solution. In general, subgrid scale techniques consist in the addition of a residual based term to the basic Galerkin formulation. The application of a standard residual based stabilization method to a linear eigenvalue problem leads to a quadratic eigenvalue problem in discrete form which is physically inconvenient. As a distinguished feature of the present study, we take the space of the unresolved subscales orthogonal to the finite element space, which promises a remedy to the above mentioned complication. In essence, we put forward that only if the orthogonal projection is used, the residual is simplified and the use of term by term stabilization is allowed. Thus, we do not need to put the whole residual in the formulation, and the linear eigenproblem form is recovered properly. We prove that the method applied is convergent, and present the error estimates for the eigenvalues and the eigenfunctions. We report several numerical tests in order to illustrate that the theoretical results are validated.Peer ReviewedPostprint (author's final draft
A stabilized finite element method for the two-field and three-field Stokes eigenvalue problems
In this paper, the stabilized finite element approximation of the Stokes
eigenvalue problems is considered for both the two-field
(displacement-pressure) and the three-field (stress-displacement-pressure)
formulations. The method presented is based on a subgrid scale concept, and
depends on the approximation of the unresolvable scales of the continuous
solution. In general, subgrid scale techniques consist in the addition of a
residual based term to the basic Galerkin formulation. The application of a
standard residual based stabilization method to a linear eigenvalue problem
leads to a quadratic eigenvalue problem in discrete form which is physically
inconvenient. As a distinguished feature of the present study, we take the
space of the unresolved subscales orthogonal to the finite element space, which
promises a remedy to the above mentioned complication. In essence, we put
forward that only if the orthogonal projection is used, the residual is
simplified and the use of term by term stabilization is allowed. Thus, we do
not need to put the whole residual in the formulation, and the linear
eigenproblem form is recovered properly. We prove that the method applied is
convergent, and present the error estimates for the eigenvalues and the
eigenfunctions. We report several numerical tests in order to illustrate that
the theoretical results are validated
Recursive integral method for transmission eigenvalues
Recently, a new eigenvalue problem, called the transmission eigenvalue
problem, has attracted many researchers. The problem arose in inverse
scattering theory for inhomogeneous media and has important applications in a
variety of inverse problems for target identification and nondestructive
testing. The problem is numerically challenging because it is non-selfadjoint
and nonlinear. In this paper, we propose a recursive integral method for
computing transmission eigenvalues from a finite element discretization of the
continuous problem. The method, which overcomes some difficulties of existing
methods, is based on eigenprojectors of compact operators. It is
self-correcting, can separate nearby eigenvalues, and does not require an
initial approximation based on some a priori spectral information. These
features make the method well suited for the transmission eigenvalue problem
whose spectrum is complicated. Numerical examples show that the method is
effective and robust.Comment: 18 pages, 8 figure
A virtual element method for the vibration problem of Kirchhoff plates
The aim of this paper is to develop a virtual element method (VEM) for the
vibration problem of thin plates on polygonal meshes. We consider a variational
formulation relying only on the transverse displacement of the plate and
propose an conforming discretization by means of the VEM which is
simple in terms of degrees of freedom and coding aspects. Under standard
assumptions on the computational domain, we establish that the resulting
schemeprovides a correct approximation of the spectrum and prove optimal order
error estimates for the eigenfunctions and a double order for the eigenvalues.
The analysis restricts to simply connected polygonal clamped plates, not
necessarily convex. Finally, we report several numerical experiments
illustrating the behaviour of the proposed scheme and confirming our
theoretical results on different families of meshes. Additional examples of
cases not covered by our theory are also presented
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