Optimal Convergence of the Original DG Method on Special Meshes for Variable Transport Velocity

Authors:
Bernardo Cockburn;Bo Dong;Johnny Guzmán;Jianliang Qian
Affiliations:
cockburn@math.umn.edu;bdong@dam.brown.edu and Johnny_Guzman@brown.edu;-;qian@math.msu.edu
Venue:
SIAM Journal on Numerical Analysis
Year:
2010

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Abstract

We prove optimal convergence rates for the approximation provided by the original discontinuous Galerkin method for the transport-reaction problem. This is achieved in any dimension on meshes related in a suitable way to the possibly variable velocity carrying out the transport. Thus, if the method uses polynomials of degree $k$, the $L^2$-norm of the error is of order $k+1$. Moreover, we also show that, by means of an element-by-element postprocessing, a new approximate flux can be obtained which superconverges with order $k+1$.