Inner products in covolume and mimetic methods
University of Richmond, Richmond, VA, USA. firstname.lastname@example.org
Revised: 29 February 2008
A class of compatible spatial discretizations for solving partial differential equations is presented. A discrete exact sequence framework is developed to classify these methods which include the mimetic and the covolume methods as well as certain low-order finite element methods. This construction ensures discrete analogs of the differential operators that satisfy the identities and theorems of vector calculus, in particular a Helmholtz decomposition theorem for the discrete function spaces. This paper demonstrates that these methods differ only in their choice of discrete inner product. Finally, certain uniqueness results for the covolume inner product are shown.
Mathematics Subject Classification: 65N06
Key words: Compatible discretization / discrete Helmholtz orthogonality / discrete exact sequence / mimetic method / covolume method.
© EDP Sciences, SMAI, 2008