Analysis and application of the IIPG method to quasilinear nonstationary convection-diffusion problems (Q955046)

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