The Resolvent Stability Condition for Spectra Convergence with Application to the Finite Element Approximation of Noncompact Operators
Wendell H. Mills · SIAM Journal on Numerical Analysis · 1979
Let $T,\{ T_n \} $ be closed linear operators on a Banach space, B, and D a compact component of the spectrum of T. Under strong convergence, $T_n \xrightarrow[s]{}T$, an applicable sufficient condition (resolvent stability around D) is developed to assure that spect $(T_n ) \to D$ as $n \to \infty $. This result is applied to the finite element method for approximating linear partial differential eigenvalue problems, $Lu = \lambda Mu$ where $L^{ - 1} M$ is assumed noncompact. The application shows the finite element spectra converges to all compact spectral components of the given operator.