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

Low-order continuous finite element spaces on hybrid non-conforming hexahedral-tetrahedral meshes