A posteriori error estimates for non-conforming approximation of eigenvalue problems

Abstract

Abstract. We consider the approximation of eigenvalue problem for the laplacian by the Crouzeix-Raviart non conforming finite elements in two and three dimensions.Extending known techniques for source problems, we introduce a posteriori error estimators for eigenvectors and eigenvalues. We prove that the error estimator is equivalent to the energy norm of the eigenvector error up to higher order terms. Moreover, we prove that our estimator provides an upper bound for the error in the approximation of the first eigenvalue, also up to higher order terms.We present numerical examples of an adaptive procedure based on our error estimator in two and three dimensions. These examples show that the error in the adaptive procedure is optimal in terms of the number of degrees of freedom.

A posteriori error analysis for nonconforming approximation of multiple eigenvalues

“…To any element , we associate an element obtained by averaging the value of v at the vertices of the triangulation . Namely, following , for each internal vertex , we consider all elements for which share the vertex P and define where w i are suitable weights such that . Lemma The following estimates hold true with constants C independent of h Proof Let be a vertex of the mesh.…”

“…In Section 3, we study the error estimates for the eigenfunctions, and, recalling the results of , we show that the results extend in a natural way to the case of multiple eigenvalues. In Section 4, using some special tools adapted from , we extend the estimates for the eigenvalues to the general case of multiplicity q ≥1.…”

Adaptive Nonconforming Finite Element Approximation of Eigenvalue Clusters

“…Let, for any subset K ⊆ T, For simple eigenvalues this type of error estimator was introduced in [29]. The adaptive algorithm is driven by this computable error estimator and runs the following loop.…”

Nonconforming finite element analysis for Poisson eigenvalue problem

“…Up to now, a lot of studies of (1) have been devoted to the finite element methods (FEMs for short). For example, [1,2,5,6] estimated the lower and upper bounds for the eigenvalues; [7,9,10] derived asymptotic expansions and extrapolations for the eigenvalues; [13] discussed a posteriori error estimates by utilizing a nonconforming element; [15][16][17][18][19] developed the adaptive and two-grid FEMs, respectively.…”

“…Eigenvalue problems play a very important role in mathematical physics and engineering technology, which have gradually gained more and more scholars' attention [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Up to now, a lot of studies of (1) have been devoted to the finite element methods (FEMs for short).…”