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Error estimates for a discontinuous Galerkin method with interior penalties applied to nonlinear Sobolev equations. (English) Zbl 1145.65063

A discontinuous Galerkin method with interior penalties is proposed for solving nonlinear Sobolev equations with evolution terms. In this sense, a symmetric semi-discrete and a family of symmetric fully-discrete time approximate schemes are formulated. \(Hp\)-version error estimates are analyzed for these schemes. For the semi-discrete time scheme an a priori \(L^{\infty}(H^{1})\) error estimate is derived and similarly, \(l^\infty \) and \(l^{2}(H^{1})\) error bounds are obtained for the fully-discrete time schemes. These results indicate that spatial rates in \(H^{1}\) and time truncation errors in \(L^{2}\) are optimal.

MSC:

65M15 Error bounds 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
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35Q30 Navier-Stokes equations
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