Essential Spectrum and Mixed Type Finite Element Method

In the error analysis of finite element method, the inf-sup condition or the uniform lifting property plays an important role. In this paper, we discuss the relationship between the uniform inf-sup condition and the essential spectrum of the operator that appears in the problem. In general, one can not expect the convergence of the finite element approximation due to the spectral pollution that stems from the inappropriate mixing of the eigen-subspaces that correspond to two distinct components of the essential spectrum. As examples of our consideration, we treat the Stokes problem, mixed approximations of elliptic problems and a structural-acoustic coupling problem. In these problems, two distinct components might appear in the essential spectrum of the corresponding operators.