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Some new classes of mixed finite element methods. (English) Zbl 0651.65078
Numerical analysis, Proc. 12th Dundee Bienn. Conf., Dundee/UK 1987, Pitman Res. Notes Math. Ser. 170, 135-156 (1988).
[For the entire collection see Zbl 0643.00021.]
Two classes of finite element methods are presented for mixed variational formulations. The methods are constructed by adding to the Galerkin method least-squares like terms which are evaluated on element interiors and include mesh-dependent coefficients. The additional terms do not upset continuity requirements of the original variational formulation because they are evaluated element-wise. Also, these are residual based contributions, and therefore, the exact solution satisfies the formulation. The methods developed may be viewed as techniques to enhance stability of the basic Galerkin method in a consistent fashion.
The methods are classified according to the nature of the governing stability conditions as 1) CBB methods (Circumventing Babuska-Brezzi condition methods) 2) SBB methods (Satisfying Babuska-Brezzi condition methods). The methodology opens up new possibilities in various problems, including structural applications.
Reviewer: I.N.Katz

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics
35J25 Boundary value problems for second-order elliptic equations