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The Discontinuous Galerkin Method for Two-Dimensional Hyperbolic Problems

Summary: The Discontinuous Galerkin Method for
Two-Dimensional Hyperbolic Problems
Part I: Superconvergence Error Analysis
Slimane Adjerid and Mahboub Baccouch
Department of Mathematics
Virginia Polytechnic Institute and State University
Blacksburg, VA 24061
February 12, 2007
In this paper we investigate the superconvergence properties of the discontin-
uous Galerkin method applied to scalar first-order hyperbolic partial differential
equations on triangular meshes. We show that the discontinuous finite element
solution is O(hp+2) superconvergent at the Legendre points on the outflow edge
for triangles having one outflow edge. For triangles having two outflow edges the
finite element error is O(hp+2) superconvergent at the end points of the inflow
edge. Several numerical simulations are performed to validate the theory. In Part
II of this work we explicitly write down a basis for the leading term of the error
and construct asymptotically correct a posteriori error estimates by solving local
hyperbolic problems with no boundary conditions on more general meshes.
Keywords: Discontinuous Galerkin method; hyperbolic problems; superconver-


Source: Adjerid, Slimane - Department of Mathematics, Virginia Tech


Collections: Mathematics