A POSTERIORI ERROR ESTIMATORS FOR MIXED APPROXIMATIONS OF EIGENVALUE PROBLEMS

Abstract

In this paper we introduce and analyze an a posteriori error estimator for the approximation of the eigenvalues and eigenvectors of a second-order elliptic problem obtained by the mixed finite element method of Raviart-Thomas of the lowest order. We define an error estimator of the residual type which can be computed locally from the approximate eigenvector and prove that the estimator is equivalent to the norm of the error in the approximation of the eigenvector up to higher order terms. The constants involved in this equivalence depend on the corresponding eigenvalue but are independent of the mesh size, provided the meshes satisfy the usual minimum angle condition. Moreover, the square root of the error in the approximation of the eigenvalue is also bounded by a constant times the estimator.

A Superconvergence result for mixed finite element approximations of the eigenvalue problem

“…From (3.39), we know that there exists supercovergence for both G h (u, p) − u h V and R h (u, p) − p h 0 for general second order elliptic eigenvalue problems by general mixed finite element methods. This is an extension of the results in [11][12][13].…”

“…In [11][12][13], a type of superconvergence between the eigenfunction approximation and its corresponding mixed finite element projection has been given for Laplace eigenvalue problems (A = I, ϕ = 0 and ρ = 1) by the lowest order Raviart-Thomas mixed finite element. For the general second order elliptic eigenvalue problem (1.1), there is no corresponding superconvergence result.…”

“…The mixed formulation for the eigenvalue problem (1.1) comes from computing the vibration modes of a fluid in a displacement formulations since using the displacement formulation for fluid is more convenient than using the pressure or potential as variable (see, e.g., [11]). …”

“…In [11][12][13], a superconvergence result between the lowest order Raviart-Thomas mixed finite element approximation and their corresponding mixed finite element projection for the Laplace eigenvalue problem with A = I, ϕ = 0 and ρ = 1 in (1.1) has been proven. Their analysis is based on the equivalence between the lowest order Raviart-Thomas approximation and the non-conforming Crouzeix-Raviart approximation for Laplace eigenvalue problem with Neumann boundary condition.…”

Adaptivity and a Posteriori Error Control for Bifurcation Problems I: The Bratu Problem

“…Despite a number of significant advances in the field, much of the research to date has focused on source problems. In the context of eigenvalue error estimation for determining whether a solution to a PDE is linearly stable or not, we mention the recent articles [13,14,20,23] for the finite element approximation of second-order self-adjoint elliptic eigenvalue problems. For related work, based on considering the eigenvalue problem as a parameter-dependent nonlinear equation, see Verfürth [27,28], for example.…”