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

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.

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 |