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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.

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 |