A Finite Element Lumped Mass Scheme for Solving Eigenvalue Problems of Circular Arches.
A Finite Element Lumped Mass Scheme for Solving Eigenvalue Problems of Circular Arches.
A full multigrid method for semilinear elliptic equation
A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as...
A hybrid finite element method to compute the free vibration frequencies of a clamped plate
A locking-free finite element method for the buckling problem of a non-homogeneous Timoshenko beam
The aim of this paper is to develop a finite element method which allows computing the buckling coefficients and modes of a non-homogeneous Timoshenko beam. Studying the spectral properties of a non-compact operator, we show that the relevant buckling coefficients correspond to isolated eigenvalues of finite multiplicity. Optimal order error estimates are proved for the eigenfunctions as well as a double order of convergence for the eigenvalues using classical abstract spectral approximation theory...
A locking-free finite element method for the buckling problem of a non-homogeneous Timoshenko beam
The aim of this paper is to develop a finite element method which allows computing the buckling coefficients and modes of a non-homogeneous Timoshenko beam. Studying the spectral properties of a non-compact operator, we show that the relevant buckling coefficients correspond to isolated eigenvalues of finite multiplicity. Optimal order error estimates are proved for the eigenfunctions as well as a double order of convergence for the eigenvalues using classical abstract spectral approximation theory...
A method of BAZLEY-FOX type for the eingenvalues of the LAPLACE operator
A modal synthesis method for the elastoacoustic vibration problem
A modal synthesis method to solve the elastoacoustic vibration problem is analyzed. A two-dimensional coupled fluid-solid system is considered; the solid is described by displacement variables, whereas displacement potential is used for the fluid. A particular modal synthesis leading to a symmetric eigenvalue problem is introduced. Finite element discretizations with lagrangian elements are considered for solving the uncoupled problems. Convergence for eigenvalues and eigenfunctions is proved, error...
A modal synthesis method for the elastoacoustic vibration problem
A modal synthesis method to solve the elastoacoustic vibration problem is analyzed. A two-dimensional coupled fluid-solid system is considered; the solid is described by displacement variables, whereas displacement potential is used for the fluid. A particular modal synthesis leading to a symmetric eigenvalue problem is introduced. Finite element discretizations with Lagrangian elements are considered for solving the uncoupled problems. Convergence for eigenvalues and eigenfunctions is proved,...
A multilevel Newton's method for eigenvalue problems
We propose a new type of multilevel method for solving eigenvalue problems based on Newton's method. With the proposed iteration method, solving an eigenvalue problem on the finest finite element space is replaced by solving a small scale eigenvalue problem in a coarse space and a sequence of augmented linear problems, derived by Newton step in the corresponding sequence of finite element spaces. This iteration scheme improves overall efficiency of the finite element method for solving eigenvalue...
A Multiscale Enrichment Procedure for Nonlinear Monotone Operators
In this paper, multiscale finite element methods (MsFEMs) and domain decomposition techniques are developed for a class of nonlinear elliptic problems with high-contrast coefficients. In the process, existing work on linear problems [Y. Efendiev, J. Galvis, R. Lazarov, S. Margenov and J. Ren, Robust two-level domain decomposition preconditioners for high-contrast anisotropic flows in multiscale media. Submitted.; Y. Efendiev, J. Galvis and X. Wu, J. Comput. Phys. 230 (2011) 937–955; J. Galvis and...
A “Natural” Norm for the Method of Characteristics Using Discontinuous Finite Elements : 2D and 3D case
We consider the numerical approximation of a first order stationary hyperbolic equation by the method of characteristics with pseudo time step k using discontinuous finite elements on a mesh . For this method, we exhibit a “natural” norm || ||h,k for which we show that the discrete variational problem is well posed and we obtain an error estimate. We show that when k goes to zero problem (resp. the || ||h,k norm) has as a limit problem (Ph) (resp. the || ||h norm) associated to the...
A nonlinear adaptative multiresolution method in finite differences with incremental unknowns
A note on the effect of numerical quadrature in finite element eigenvalue approximation.
A numerical study of eigenvalues of the hyperbolic Laplacian for polyhedra with one cusp.
A posteriori error estimates for a discontinuous Galerkin approximation of Steklov eigenvalue problems
We derive a residual-based a posteriori error estimator for a discontinuous Galerkin approximation of the Steklov eigenvalue problem. Moreover, we prove the reliability and efficiency of the error estimator. Numerical results are provided to verify our theoretical findings.
A proof of monotony of the Temple quotients in eigenvalue problems
If the so-called Collatz method is applied to get twosided estimates of the first eigenvalue , the sequences of the so-called Schwarz quatients (which are upper bounds for ) and of the so-called Temple quotients (which are lower bounds) are constructed. While monotony of the first sequence was proved many years ago, monotony of the second one has been proved only recently by F. goerisch and J. Albrecht in their common paper “Die Monotonie der Templeschen Quotienten” (ZAMM, in print). In the present...
A special finite element method based on component mode synthesis
The goal of our paper is to introduce basis functions for the finite element discretization of a second order linear elliptic operator with rough or highly oscillating coefficients. The proposed basis functions are inspired by the classic idea of component mode synthesis and exploit an orthogonal decomposition of the trial subspace to minimize the energy. Numerical experiments illustrate the effectiveness of the proposed basis functions.
A Superconvergence result for mixed finite element approximations of the eigenvalue problem
In this paper, we present a superconvergence result for the mixed finite element approximations of general second order elliptic eigenvalue problems. It is known that a superconvergence result has been given by Durán et al. [Math. Models Methods Appl. Sci. 9 (1999) 1165–1178] and Gardini [ESAIM: M2AN 43 (2009) 853–865] for the lowest order Raviart-Thomas approximation of Laplace eigenvalue problems. In this work, we introduce a new way to derive the superconvergence of general second order elliptic...
A Superconvergence result for mixed finite element approximations of the eigenvalue problem∗
In this paper, we present a superconvergence result for the mixed finite element approximations of general second order elliptic eigenvalue problems. It is known that a superconvergence result has been given by Durán et al. [Math. Models Methods Appl. Sci. 9 (1999) 1165–1178] and Gardini [ESAIM: M2AN 43 (2009) 853–865] for the lowest order Raviart-Thomas approximation of Laplace eigenvalue problems. In this work, we introduce a new way to derive the superconvergence of general second order elliptic...