A posteriori error estimates for eigenvalue problems in mixed form

Francesca Gardini

In this talk we present reliable and efficient a posteriori error estimators for both Brezzi-Douglas-Marini (BDM) and Raviart-Thomas (RT) approximation of eigenvalue problems.

According to the author's knowledge, it is the first time that an a posteriori error analysis for mixed approximation of eigenvalue problems is developed without using the equivalence with a non conforming finite element approximation.

The problem we deal with arises from the displacement formulation to compute the vibration modes of an acoustic fluid contained within a rigid cavity. In two dimensions this problem is equivalent to Maxwell's eigenproblem for a cavity resonator with dielectric and permeability constants equal to 1.

We first consider BDM approximation and define an error estimator of the residual type. Assuming the smoothness of the solution, we prove that the error estimator is equivalent to the H(div)-norm of the error up to higher order terms. Moreover, we prove that the square root of the error in the approximation of the eigenvalue is also bounded by the error estimator. The constants involved in the above inequalities depend on the eigenvalue, but are independent of the mesh size.

We then change to RT approximation and develop a different a posteriori error analysis. We present a residual type error estimator and prove that it is equivalent to the L2-norm of the error, provided a superconvergence result holds. We also prove that the superconvergence property holds for the lowest order RT elements. Moreover, we will show the results of some numerical computations which confirm the superconvergence property for the lowest order RT space and suggest that the superconvergence property holds for the lowest order BDM space as well.

Eventually, the results of some preliminaries numerical tests which show the efficiency of the error indicators when used to point out which elements have to be refined will be also presented.