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A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. (English) Zbl 0729.65085
This note gives a numerical illustration of some convergence results got in the approximation of the first-order hyperbolic equation \(\alpha \cdot \nabla u+\beta u=f\) in \(\Omega\), \(u=g\) on \(\partial \Omega\), by the discontinuous Galerkin method, where \(\Omega\) is a bounded polygonal domain in \({\mathbb{R}}^ 2\) and \(\partial \Omega\) its inflow boundary, \(\alpha\) is a constant unit vector, and \(\beta\) is a bounded function. This numerical example shows that the known \(L^ 2\) error estimate \(\| u-u^ h\| \leq Ch^{k+1/2}\| u\|_{k+1}\) cannot be improved within the class of quasi-uniform meshes, not even for smooth exact solution u.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35L50 Initial-boundary value problems for first-order hyperbolic systems
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