Mixed finite elements and the complex of Withney forms. (English) Zbl 0692.65053

Bossavit, A.

The mathematics of finite elements and applications VI, MAFELAP 1987, Proc. 6th Conf., Uxbridge/UK 1987, 137-144 (1988).

[For the entire collection see Zbl 0652.00018.]
This paper is devoted to the numerical solution of partial differential equations in three (or less) dimensions. The technique of mixed finite elements is employed. Main result: In so called “mixed” formulations one has two unknown fields and the equations may look like: (1) \(a(u,v)+b(u,v)=0\) \(\forall v\in U(\Omega)\) and (2) \(b(u,w)=(f,w)\) \(\forall w\in V(\Omega)\), (U(\(\Omega)\) and V(\(\Omega)\) are distinct Hilbert spaces). Some compatibility condition between \(U_ h(\Omega)\) and \(V_ h(\Omega)\) is needed to ensure convergence.
The author presents a different approach to mixed elements, which stems from a reformulation of (1), (2) in terms of differential forms (Whitney forms - which are for differential forms what finite elements are for scalar or vector fields). Another advantage of this approach is that no effort is needed to enforce the compatibility condition (which is automatically satisfied). Applications to some classical problems (heat, Stokes, Maxwell at low frequency) are suggested.

Reviewer: J.Lovíšek

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65Z05	Applications to the sciences
76M99	Basic methods in fluid mechanics
35J25	Boundary value problems for second-order elliptic equations
35Q99	Partial differential equations of mathematical physics and other areas of application
78M99	Basic methods for problems in optics and electromagnetic theory

Keywords:

heat equation; Stokes equation; Maxwell equation; mixed finite elements; “mixed” formulations; compatibility condition; Whitney forms; differential forms

Citations:

Zbl 0652.00018