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

Dolejší, Vít

J. Comput. Appl. Math. 222, No. 2, 251-273 (2008).

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 \[ \vec{R}:=\nu(|\nabla u|)\nabla u) \quad \text{with } \nu(w):=\nu_\infty+\frac{\nu_0-\nu_\infty}{(1+w)^\gamma}, \quad \gamma>0. \]

Reviewer: Rolf Dieter Grigorieff (Berlin)

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

Keywords:

quasilinear convection-diffusion equations; discontinuous Galerkin finite element method; incomplete interior penalty Galerkin method; method of lines; error estimates; convergence; viscous compressible flows; numerical experiments

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