Existence and stability, and discrete BB and rank conditions, for general mixed-hybrid finite elements in elasticity. (English) Zbl 0624.73094

Hybrid and mixed finite element methods, Winter Annu. Meet. ASME, Miami Beach/Fl. 1985, AMD-Vols. 73, 91-112 (1985).
[For the entire collection see Zbl 0623.00027.]
In this paper, all possible forms of mixed-hybrid finite element methods that are based on multi-field variational principles are examined as to the conditions for existence, stability, and uniqueness of their solutions. The reasons as to why certain “simplified hybrid-mixed methods” in general, and the so-called “simplified hybrid-displacement method” in particular (based on the so-called simplified variational principles), become unstable, are discussed. A comprehensive discussion of the “discrete” BB-conditions, and the rank conditions, of the matrices arising in mixed-hybrid methods, is given. Some recent studies aimed in the assurance of such rank conditions, and the related problem of the avoidance of spurious kinematic modes, are presented.


74S05 Finite element methods applied to problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
65K10 Numerical optimization and variational techniques


Zbl 0623.00027