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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 ${L}^{\infty }\left({H}^{1}\right)$ error estimate is derived and similarly, ${l}^{\infty }$ and ${l}^{2}\left({H}^{1}\right)$ 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 (IVP of PDE) 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) 65M20 Method of lines (IVP of PDE) 35Q30 Stokes and Navier-Stokes equations