## 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

Zbl 0618.65105
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### References:

 [1] Richard S. Falk and Gerard R. Richter, Analysis of a continuous finite element method for hyperbolic equations, SIAM J. Numer. Anal. 24 (1987), no. 2, 257 – 278. · Zbl 0619.65100 [2] C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp. 46 (1986), no. 173, 1 – 26. · Zbl 0618.65105 [3] Claes Johnson and Juhani Pitkäranta, Convergence of a fully discrete scheme for two-dimensional neutron transport, SIAM J. Numer. Anal. 20 (1983), no. 5, 951 – 966. · Zbl 0538.65097 [4] P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 89 – 123. Publication No. 33. · Zbl 0341.65076 [5] W. H. Reed & T. R. Hill, Triangular Mesh Methods for the Neutron Transport Equation, Los Alamos Scientific Laboratory Report LA-UR-73-479.
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