Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes (Q2877390)

The authors present an analysis for compatible discrete operator schemes for elliptic problems on polyhedral meshes. One of the main tools in the construction of the scheme is the discrete Hodge operator linking gradients to fluxes by means of a dual mesh, while a structure-preserving discretization is employed for the gradient and divergence operators. The discrete Hodge operator is sparse, symmetric positive definite and is assembled cellwise from local operators. Some new functional analysis results on the discrete gradient that are the counterpart of the Sobolev embeddings are derived. The two design properties of the local discrete Hodge operators yielding optimal discrete energy error estimates are identified. The authors also show how these operators can be built from local nonconforming gradient reconstructions using a dual barycentric mesh. An optimal \(L^ 2\)-error estimate for the potential for smooth solutions is proved. Links with existing schemes (finite elements, finite volumes, mimetic finite differences) are discussed. Numerical results are presented on three-dimensional polyhedral meshes.