## Analysis and application of the IIPG method to quasilinear nonstationary convection-diffusion problems.(English)Zbl 1165.65055

The author considers the following nonstationary quasilinear convection diffusion problem
$\frac{\partial u}{\partial t}+\nabla \cdot \vec{f}(u)=\nabla \cdot \vec{R}(u,\nabla u)+g \quad \text{ in } \Omega \times (0,T)$
where $$\Omega \subset \mathbb{R}^d, d=2,3$$, subject to initial- and mixed Dirichlet/Neumann boundary conditions. The vector function $$\vec{f}$$ is assumed to be globally Lipschitz and the components $$R_s$$ in the diffusion term $$\vec{R}$$ to be linearly bounded. Moreover, the Jacobian of $$\vec{R}$$ must satisfy a monotonicity condition.
The problem is discretized with the incomplete interior penalty Galerkin (IIPG) method using polynomial finite elements with not too much varying order between neighbouring elements. The main result is an $$hp$$ a priori error estimate for the method of lines. Numerical experiments are presented, where $\vec{R}:=\nu(|\nabla u|)\nabla u) \quad \text{with } \nu(w):=\nu_\infty+\frac{\nu_0-\nu_\infty}{(1+w)^\gamma}, \quad \gamma>0.$

### MSC:

 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 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 35K55 Nonlinear parabolic equations
Full Text:

### References:

  Arnold, D.N., An interior penalty finite element method with discontinuous elements, SIAM J. numer. anal., 19, 4, 742-760, (1982) · Zbl 0482.65060  Arnold, D.N.; Brezzi, F.; Cockburn, B.; Marini, L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. numer. anal., 39, 5, 1749-1779, (2002) · Zbl 1008.65080  Babuška, I.; Strouboulis, T., The finite element method and its reliability, (2001), Clarendon Press Oxford · Zbl 0997.74069  Babuška, I.; Suri, M., The $$h p$$-version of the finite element method with quasiuniform meshes, M^{2}AN math. model. numer. anal., 21, 199-238, (1987) · Zbl 0623.65113  Babuška, I.; Suri, M., The $$p$$- and $$h$$-$$p$$ versions of the finite element method. an overview, Comput. methods appl. mech. engrg., 80, 5-26, (1990) · Zbl 0731.73078  Ciarlet, P.G., The finite elements method for elliptic problems, (1979), North-Holland Amsterdam, New York, Oxford  Cockburn, B., Discontinuous Galerkin methods for convection dominated problems, (), 69-224 · Zbl 0937.76049  Discontinuous Galerkin methods. theory computation and applications, ()  Dawson, C.N.; Sun, S.; Wheeler, M.F., Compatible algorithms for coupled flow and transport, Comput. methods appl. mech. engrg., 193, 2565-2580, (2004) · Zbl 1067.76565  Dolejší, V., Higher order semi-implicit discontinuous Galerkin finite element schemes for nonlinear convection – diffusion problems, () · Zbl 1119.65387  V. Dolejší, Semi-implicit interior penalty discontinuous Galerkin methods for viscous compressible flows, Commun. Comput. Phys  V. Dolejší, M. Feistauer, V. Kučera, V. Sobotíková, An optimal $$L^\infty(L^2)$$-error estimate of the discontinuous Galerkin method for a nonlinear nonstationary convection – diffusion problem, IMA J. Numer. Anal. Preprint No. MATH-knm-2007/1, Charles University Prague, School of Mathematics. www.karlin.mff.cuni.cz/ms-preprints, 2007  Dolejší, V.; Feistauer, M.; Schwab, C., A finite volume discontinuous Galerkin scheme for nonlinear convection – diffusion problems, Calcolo, 39, 1-40, (2002) · Zbl 1098.65095  Dolejší, V.; Feistauer, M.; Sobotíková, V., A discontinuous Galerkin method for nonlinear convection – diffusion problems, Comput. methods appl. mech. engrg., 194, 2709-2733, (2005) · Zbl 1093.76034  M. Feistauer, On the finite element approximation of functions with noninteger derivatives, Numer. Funct. Anal. Optim. 10, 91-110 · Zbl 0668.65008  Feistauer, M., Mathematical methods in fluid dynamics, (1993), Longman Scientific & Technical Harlow · Zbl 0819.76001  Feistauer, M.; Felcman, J.; Straškraba, I., Mathematical and computational methods for compressible flow, (2003), Oxford University Press Oxford · Zbl 1028.76001  Feistauer, M.; Ženísek, A., Finite element solution of nonlinear elliptic problems, Numer. math., 50, 451-475, (1987) · Zbl 0637.65107  Houston, P.; Robson, J.; Süli, E., Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: the scalar case, IMA J. numer. anal., 25, 726-749, (2005) · Zbl 1084.65116  Houston, P.; Schwab, C.; Süli, E., Discontinuous $$h p$$-finite element methods for advection – diffusion problems, SIAM J. numer. anal., 39, 6, 2133-2163, (2002) · Zbl 1015.65067  Rivière, B.; Wheeler, M.F., A discontinuous Galerkin method applied to nonlinear parabolic equations, (), 231-244 · Zbl 0946.65078  Rivière, B.; Wheeler, M.F.; Girault, V., Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. i, Comput. geosci., 3, 3-4, 337-360, (1999) · Zbl 0951.65108  Schwab, C., $$p$$- and $$h p$$-finite element methods, (1998), Clarendon Press Oxford  S. Sun, Discontinuous Galerkin methods for reactive transport in porous media, Ph.D. Thesis, The University of Texas, Austin, 2003  Sun, S.; Wheeler, M.F., Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media, SIAM J. numer. anal., 43, 1, 195-219, (2005) · Zbl 1086.76043  Temam, R., Navier – stokes equations. theory and numerical analysis, (1977), North-Holland Amsterdam, New York, Oxford · Zbl 0383.35057  Wesseling, P., Principles of computational fluid dynamics, (2001), Springer Berlin · Zbl 0989.76069  Zeidler, E., Nonlinear functional analysis and its applications. II/B, nonlinear monotone operators, (1985), Springer New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.