The Cell Method: quadratic interpolation with tetrahedra for 3D scalar fields

The Cell Method (CM) is a numerical method to solve field equations starting from its direct algebraic formulation. For two-dimensional problems it has been demonstrated that using simplicial elements with an affine interpolation, the CM obtains the same fundamental equation of the Finite Element Method (FEM); using the quadratic interpolation functions, the fundamental equation differs depending on how the dual cell is defined. In spite of that, the CM can still provide the same convergence rate obtainable with the FEM. Particularly, adopting a uniform triangulation and basing the dual cells on the Gauss points of the primal edges, the CM is able to reach the 4th order of convergence.

In this note the use of quadratic interpolation to solve the Laplace equation in three dimensional problems is presented, adopting as primal cell the 10-nodes tetrahedron. A convergence analysis demonstrates that, as in two-dimensional case, the CM with quadratic interpolation can obtain the fourth order of convergence in solving the Laplace equation.