  This article deals with solving partial differential equations with the
finite element method on hybrid non-conforming hexahedral-tetrahedral meshes.
By non-conforming, we mean that a quadrangular face of a hexahedron can be
connected to two triangular faces of tetrahedra. We introduce a set of
low-order continuous (C0) finite element spaces defined on these meshes. They
are built from standard tri-linear and quadratic Lagrange finite elements with
an extra set of constraints at non-conforming hexahedra-tetrahedra junctions to
recover continuity. We consider both the continuity of the geometry and the
continuity of the function basis as follows: the continuity of the geometry is
achieved by using quadratic mappings for tetrahedra connected to tri-affine
hexahedra and the continuity of interpolating functions is enforced in a
similar manner by using quadratic Lagrange basis on tetrahedra with constraints
at non-conforming junctions to match tri-linear hexahedra. The so-defined
function spaces are validated numerically on simple Poisson and linear
elasticity problems for which an analytical solution is known. We observe that
using a hybrid mesh with the proposed function spaces results in an accuracy
significantly better than when using linear tetrahedra and slightly worse than
when solely using tri-linear hexahedra. As a consequence, the proposed function
spaces may be a promising alternative for complex geometries that are out of
reach of existing full hexahedral meshing methods.
