## Section: New Results

### Finite elements on quadrilateral and hexahedral meshes

Participants : Roland Becker, Robert Luce, David Trujillo.

The construction of finite element methods on quadrilateral, and particularly, hexahedral meshes can be a complicated task; especially the development of mixed and non-conforming methods is an active field of research. The difficulties arise not only from the fact that adequate degrees of freedom have to be found, but also from the non-constantness of the element Jacobians; an arbitrary hexahedron, which we define as the image of the unit cube under a tri-linear transformation, does in general not have plane faces, which implies for example, that the normal vector is not constant on a side.

We have built a new class of finite element functions (named pseudo-conforming) on quadrilateral and hexahedral meshes. The degrees of freedom are the same as those of classical iso-parametric finite elements but the basis functions are defined as polynomials on each element of the mesh. On general quadrilaterals and hexahedra, our method leads to a non-conforming method; in the particular case of parallelotopes, the new finite elements coincide with the classical ones [16] , [15] . This approach is a first step towards higher-order methods on arbitrary hexahedral meshes, see Section 3.6 .

A special feature of these meshes is the possibility of relatively simple hierarchical local refinement, under the condition that hanging nodes are introduced. We have analyzed such an adaptive methods on quadrilateral meshes [44] , see Section 6.1 .