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Error estimates to smooth solutions of semi-discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws. (English) Zbl 1361.65067
This paper deals with error estimates to smooth solutions of semi-discrete discontinuous Galerkin (DG) methods with the $$P^k$$ finite element space of piecewise $$k$$th degree polynomials and quadrature rules for scalar conservation laws. In the case of 1D and under the assumption that the exact solution and the physical flux are smooth, it is shown that the error estimate is of order $$O(h^{k+\frac{1}{2}})$$ for general monotone fluxes while the order is $$O(h^{k+1})$$ for upwind fluxes if the quadrature over elements is exact for polynomials of degree $$(2k)$$. For multidimensional problems, it is proved that, if the quadrature over elements is exact for polynomials of degree $$(2k+1)$$, the error estimate is of order $$O(h^{k+\frac{1}{2}})$$ for smooth numerical fluxes. The techniques presented are based on the use of the energy estimate and Taylor expansions first introduced by Q. Zhang and C.-W. Shu [SIAM J. Numer. Anal. 42, No. 2, 641–666 (2004; Zbl 1078.65080)]. Some numerical results are presented to support the theoretical results.

##### MSC:
 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations
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