A stabilized finite element method for the two-field and three-field Stokes eigenvalue problems

Abstract

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

Chebyshev spectral collocation method approximations of the Stokes eigenvalue problem based on penalty techniques

“…The tabulated values show that the approximations obtained from the classical penalisation agree reasonably well with the reference values. Further, the plot of the eigenfunction associated to the minimum eigenvalue is given in Figure 3, revealing a good agreement with the existing ones in the literature (see, [17]). In order to investigate the convergence behaviour of the approximated eigenvalue with respect to the penalty parameter, the minimum eigenvalue (denoted by λ 1 ) is calculated for ε values varying from 10 −1 to 10 −10 .…”

Modal analysis of elastic vibrations of incompressible materials using a pressure-stabilized finite element method

“…In computing the modal expansion of the solutions, we depend on the accuracy of the approximated eigensolutions. As already mentioned, it has been established in [38] that the discrete eigensolutions convergence optimally to their continuous counterparts. Consequently, to determine the number of elements used in the simulations, for each problem we have carried out a mesh test in which we have examined the maximum relative change in the first 10 frequencies contributing into expansion (46) between two successive mesh levels.…”

“…Note that the last term disappears if Q h is made of continuous functions. This is the two-field formulation for the Stokes EVP considered in [38]. It is shown there that it provides stable and optimally accurate eigenfunctions and eigenvalues.…”

“…A residual based stabilization technique may lead to a quadratic EVP even if it is applied to approximate a linear EVP. We have proposed a FEM for the Stokes EVP that preserves the linearity of the continuous problem in [38]. The method is framed within the variational multiscale (VMS) concept, which assumes that the unknown can be split into a finite element component and a subgrid scale that needs to be modeled, and it has been applied to solve stationary boundary value problems in incompressible linear elasticity models in [11][12][13]28], for example.…”

Arnold--Winther Mixed Finite Elements for Stokes Eigenvalue Problems

“…In [21], Meddahi et al proposed a finite element analysis of a pseudo-stress formulation for the Stokes eigenvalue problem. In [25], Türk et al introduced a stabilized finite element method for two-field and three-field Stokes eigenvalue problems. From [22], one can also study a variety of mixed or hybrid finite element methods for eigenvalue problems.…”