Higher-order mimetic methods for unstructured meshes

A higher-order mimetic method for the solution of partial differential equations on unstructured meshes is developed and demonstrated on the problem of conductive heat transfer. Mimetic discretization methods create discrete versions of the partial differential operators (such and the gradient and divergence) that are exact in some sense and therefore mimic the important mathematical properties of their continuous counterparts. The proposed numerical method is an interesting mixture of both finite volume and finite element ideas. While the ideas presented can be applied to arbitrarily high-order accuracy, we focus in this work on the details of creating a third-order accurate method. The proposed method is shown to be exact for piecewise quadratic solutions and shows third-order convergence on arbitrary triangular/tetrahedral meshes. The numerical accuracy of the method is confirmed on both two-dimensional and three-dimensional unstructured meshes. The computational cost required for a desired accuracy is analyzed against lower-order mimetic methods.