Compatible Discretizations for Eigenvalue Problems
Saturday, May 15, 2004 - 11:00am - 11:50am
We start with a short review of standard Galerkin approximation of variationally posed eigenvalue problems, where we mainly consider the case of elliptic problems associated with a compact resolvent. Our basic example is the Laplace/Poisson eigenvalue problem. The main result, in this case, can be summarized by the claim that any choice of discrete space sequence, which provides a convergent scheme for the source problem, automatically performs well when applied to the corresponding eigenvalue problem. The main discussion in this talk focuses on the approximation of eigenvalue problems in mixed form. Using again the basic example of the Laplace/Poisson eigenvalue problem, we show that, when using a mixed method for its discretization, the picture is somewhat different from the previous case. The main (counter) example is given by a choice of discrete space sequences such that the classical Brezzi's conditions are satisfied (whence the source problem is correctly approximated) but, when the eigenvalue problem is considered, several spurious eigenmodes pollute the discrete spectrum. This surprising behavior is proved theoretically and numerically demonstrated. We then review the theory of the discretization of eigenvalue problems in mixed form (joint work with F. Brezzi and L. Gastaldi), where the compatibilities between the discrete space sequences for the good approximation of eigenpairs are made clear. Necessary and sufficient conditions are given for several general cases of interest. If time permits, some consequences of the theory can be presented. One application of the theory is the approximation of interior Maxwell's eigenvalue (leading in particular to the discretization of time-harmonic Maxwell's system). The results can be suitably modified to include the approximation of band gaps for photonic crystals (joint work with M. Conforti and L. Gastaldi). Another important consequence concerns the approximation to evolution problems in mixed form (joint work with L. Gastaldi).