The Continuous Galerkin Method Is Locally Conservative

Abstract

We examine the conservation law structure of the continuous Galerkin method. We employ the scalar, advection–diffusion equation as a model problem for this purpose, but our results are quite general and apply to time-dependent, nonlinear systems as well. In addition to global conservation laws, we establish local conservation laws which pertain to subdomains consisting of a union of elements as well as individual elements. These results are somewhat surprising and contradict the widely held opinion that the continuous Galerkin method is not locally conservative.