## 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
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### References:

 [1] Arnold, D.N., An interior penalty finite element method with discontinuous elements, SIAM J. numer. anal., 19, 4, 742-760, (1982) · Zbl 0482.65060 [2] 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 [3] Babuška, I.; Strouboulis, T., The finite element method and its reliability, (2001), Clarendon Press Oxford · Zbl 0997.74069 [4] 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 [5] 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 [6] Ciarlet, P.G., The finite elements method for elliptic problems, (1979), North-Holland Amsterdam, New York, Oxford [7] Cockburn, B., Discontinuous Galerkin methods for convection dominated problems, (), 69-224 · Zbl 0937.76049 [8] Discontinuous Galerkin methods. theory computation and applications, () [9] 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 [10] Dolejší, V., Higher order semi-implicit discontinuous Galerkin finite element schemes for nonlinear convection – diffusion problems, () · Zbl 1119.65387 [11] V. Dolejší, Semi-implicit interior penalty discontinuous Galerkin methods for viscous compressible flows, Commun. Comput. Phys [12] 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 [13] 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 [14] 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 [15] M. Feistauer, On the finite element approximation of functions with noninteger derivatives, Numer. Funct. Anal. Optim. 10, 91-110 · Zbl 0668.65008 [16] Feistauer, M., Mathematical methods in fluid dynamics, (1993), Longman Scientific & Technical Harlow · Zbl 0819.76001 [17] Feistauer, M.; Felcman, J.; Straškraba, I., Mathematical and computational methods for compressible flow, (2003), Oxford University Press Oxford · Zbl 1028.76001 [18] Feistauer, M.; Ženísek, A., Finite element solution of nonlinear elliptic problems, Numer. math., 50, 451-475, (1987) · Zbl 0637.65107 [19] 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 [20] 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 [21] Rivière, B.; Wheeler, M.F., A discontinuous Galerkin method applied to nonlinear parabolic equations, (), 231-244 · Zbl 0946.65078 [22] 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 [23] Schwab, C., $$p$$- and $$h p$$-finite element methods, (1998), Clarendon Press Oxford [24] S. Sun, Discontinuous Galerkin methods for reactive transport in porous media, Ph.D. Thesis, The University of Texas, Austin, 2003 [25] 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 [26] Temam, R., Navier – stokes equations. theory and numerical analysis, (1977), North-Holland Amsterdam, New York, Oxford · Zbl 0383.35057 [27] Wesseling, P., Principles of computational fluid dynamics, (2001), Springer Berlin · Zbl 0989.76069 [28] Zeidler, E., Nonlinear functional analysis and its applications. II/B, nonlinear monotone operators, (1985), Springer New York
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