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An optimal-order error estimate for the discontinuous Galerkin method. (English) Zbl 0643.65059
A discontinuous Galerkin method based on polynomials of degree n is analyzed for the partial differential equation \(au_ x+bu_ y=f\) on a domain \(\Omega\) in the plane. Initial values of u are specified in \(\Omega\), and boundary data are given on the inflow portion of the boundary. The triangulation is regular in strips. If h is the diameter of the largest triangle, it is shown that the error is of the order of \(h^{n+1}\) if the solution u is in the Sobolev space \(H_{n+2}(\Omega)\). This estimate is an improvement by \(h^{1/2}\) over a result of C. Johnson and J. Pitkäranta [ibid. 46, 1-26 (1986; Zbl 0618.65105)].
Reviewer: G.Hedstrom

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