Two classes of mixed finite element methods

Finite element methods are presented in an abstract settings for mixed variational formulations. The methods are constructed by adding to the classical Galerkin method various least-squares like terms. The additional terms involve integrals over element interiors, and include mesh-parameter dependent coefficients. The methods are designed to enhance stability. Consistency is achieved in the sense that exact solutions identically satisfy the variational equations. Applied to various problems, simple finite element interpolations are rendered convergent, including convenient equal-order interpolations which are generally unstable within the Galerkin approach. The methods are subdivided into two classes according to the manner in which stability is attained: (1) Circumventing Babuška-Brezzi condition methods. (2) Satisfying Babuška-Brezzi condition methods. Convergence is established for each class of methods. Applications of the first class of methods to Stokes flow and compressible linear elasticity are presented. The second class of methods is applied to compressible and incompressible elasticity problems.