Spurious Solutions in the Approximation of Eigenproblems of Mixed Form

Daniele Boffi
UniversitÓ di Pavia and PSU

Abstract: As a sample problem we consider the approximation of Laplace eigenvalues by means of mixed finite elements. On a suitable criss-cross mesh we construct a pair of spaces satisfying the standard hypotheses for the stability of mixed approximations (ellipticity in the kernel and inf-sup condition), but we show that this choice of spaces cannot avoid the presence of spurious discrete solutions. We present an abstract setting for the discretization of the eigenvalues of linear elliptic problems in mixed form, giving necessary and sufficient conditions for the convergence. In this framework we can easily prove, for instance, that the well-known Raviart-Thomas element gives good results when applied to this problem.