Mixed methods on quadrilaterals and hexahedra
The author deals with mixed finite element methods on quadrilaterals and hexahedra and describes a new family of discrete spaces suitable for their use. The new spaces are natural in the sense of differential geometry, so all the usual mixed method theory, including hybrid formulation, carries over to these new elements with proofs unchanged. These spaces have good approximation properties since they are just familiar spaces of polynomials, in the transformed coordinate system. Finite element methods are poorly suited to certain porous media computations because they do not conserve mass element by element. Thus the author likes to use mixed methods on quadrilateral elements as they do conserve mass locally as well as globally. Mixed methods have generally been applied to parallelograms, triangles and other shapes which are affine transformations of reference elements. Because transforming quadrilaterals to squares is a non-affine map, and because mixed methods involve divergence computations, which are not the same in all coordinate systems, the natural generalization of mixed methods to quadrilaterals is somewhat more complicated than either the corresponding Raviart-Thomas spaces for rectangles or corresponding finite element spaces for quadrilaterals. On use of bilinear mapping, the resulting square reference element can be viewed as a manifold with an induced metric which is different from the usual Euclidean metric. This has a number of consequences. The outward normal can vary along edges. Distinction must be made between covariant and contravariant vector components. More significantly the divergence of the vector field is no longer equal to the trace of its Jacobian matrix. As a result the usual Raviart-Thomas spaces no longer satisfy the compatibility conditions. Thus arises the necessity of introducing new spaces which are compatible. Degeneracy in the new spaces is avoided in order to prevent conditioning difficulties with nearly rectangular elements and good approximation properties are built in. Unfortunately nonlinearities turn out to limit the applicability of the new spaces to meshes obtained from a rectangular mesh through the application of a single global transformation. Alternative approaches defining mixed methods on quadrilaterals are possible which remove the restriction on the mesh. Details regarding this are given at the end of the paper.
scientific article; zbMATH DE number 640746