Gershgorin theory for stiffness and stability of evolution systems and convection-diffusion

An analysis of stiffness and stability based on Gershgorin theorems for eigenvalues is developed for initial value systems. In particular, semidiscrete formulations for evolution problems are analysed. Common techniques such as semidiscrete finite difference and finite element methods are examined using eigenvalue bounds to characterize stiffness and stability of the associated systems. The analysis is applied to a prototype convection-diffusion problem to demonstrate the arguments and clarify several current questions concerning the qualitative nature of the solution and errors, including effects of: “lumped” versus “consistent” finite element formulations; high- or low-degree bases; mesh refinement, dimensionality and differing material properties. To study general initial value systems such as those arising in consistent finite element formulations, a generalized Gershgorin theory and computable bounds in the chordal metric are utilized.