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.

Classification: 65N06

Keywords: compatible discretization, discrete Helmholtz orthogonality, discrete exact sequence, mimetic method, covolume method

@article{M2AN_2008__42_6_941_0, author = {Trapp, Kathryn A.}, title = {Inner products in covolume and mimetic methods}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {42}, number = {6}, year = {2008}, pages = {941-959}, doi = {10.1051/m2an:2008030}, zbl = {1155.65103}, mrnumber = {2473315}, language = {en}, url = {http://www.numdam.org/item/M2AN_2008__42_6_941_0} }

Trapp, Kathryn A. Inner products in covolume and mimetic methods. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 6, pp. 941-959. doi : 10.1051/m2an:2008030. http://www.numdam.org/item/M2AN_2008__42_6_941_0/

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