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Inner products in covolume and mimetic methods

Published online by Cambridge University Press:  30 July 2008

Kathryn A. Trapp
Kathryn A. Trapp*
Affiliation:
University of Richmond, Richmond, VA, USA. ktrapp@richmond.edu
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Abstract

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.

Keywords

Compatible discretizationdiscrete Helmholtz orthogonalitydiscrete exact sequencemimetic methodcovolume method.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 6 , November 2008 , pp. 941 - 959
Copyright
© EDP Sciences, SMAI, 2008

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