The enriched Crouzeix-Raviart elements are equivalent to the Raviart-Thomas elements

For both the Poisson model problem and the Stokes problem in any dimension, this paper proves that the enriched Crouzeix-Raviart elements are actually identical to the first order Raviart-Thomas elements in the sense that they produce the same discrete stresses. This result improves the previous result in literature which, for two dimensions, states that the piecewise constant projection of the stress by the first order Raviart-Thomas element is equal to that by the Crouzeix-Raviart element. For the eigenvalue problem of Laplace operator, this paper proves that the error of the enriched Crouzeix-Raviart element is equivalent to that of the Raviart-Thomas element up to higher order terms

Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods

This article is devoted to computing the lower and upper bounds of the Laplace eigenvalue problem. By using the special nonconforming finite elements, i.e., enriched Crouzeix-Raviart element and extension $Q_1^{\rm rot}$, we get the lower bound of the eigenvalue. Additionally, we also use conforming finite elements to do the postprocessing to get the upper bound of the eigenvalue. The postprocessing method need only to solve the corresponding source problems and a small eigenvalue problem if higher order postprocessing method is implemented. Thus, we can obtain the lower and upper bounds of the eigenvalues simultaneously by solving eigenvalue problem only once. Some numerical results are also presented to validate our theoretical analysis.Comment: 19 pages, 4 figure

Fully computable a posteriori error bounds for eigenfunctions

Fully computable a posteriori error estimates for eigenfunctions of compact self-adjoint operators in Hilbert spaces are derived. The problem of ill-conditioning of eigenfunctions in case of tight clusters and multiple eigenvalues is solved by estimating the directed distance between the spaces of exact and approximate eigenfunctions. Derived upper bounds apply to various types of eigenvalue problems, e.g. to the (generalized) matrix, Laplace, and Steklov eigenvalue problems. These bounds are suitable for arbitrary conforming approximations of eigenfunctions, and they are fully computable in terms of approximate eigenfunctions and two-sided bounds of eigenvalues. Numerical examples illustrate the efficiency of the derived error bounds for eigenfunctions.Comment: 27 pages, 8 tables, 9 figure