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The continuous Galerkin method is locally conservative. (English) Zbl 0969.65104
The authors examine the conservation law structure of the continuous Galerkin method for the scalar advection-diffusion equation, establish local conservation laws which pertain to subdomains consisting of a union of elements as well as individual elements. The results are quite general and apply to time-dependent, nonlinear systems as well and are somewhat surprising and contradict the widely held opinion that the continuous Galerkin method is not locally conservative.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35K55 Nonlinear parabolic equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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