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Analysis of a BDF-DGFE scheme for nonlinear convection-diffusion problems. (English) Zbl 1158.65068
A numerical procedure is designed to get approximate solutions of a scalar nonlinear convection-diffusion equation on a bounded polyhedral domain \(\Omega \in \mathbb{R}^d\) (\(d=2,3\)). It consists of a spatial discretization based on the symmetric variant of the discontinuous Galerkin finite element (DGFE) method and a semi-implicit \(k\)-step backward difference formula (BDF) for integrating the resulting system of stiff ordinary differential equations (ODEs) in time. The space semi-discretization algorithm, previously analysed by the authors, gives optimal order of convergence if certain technical requirements are satisfied. With respect to the discretization of the resulting ODEs in time, the so-called implicit-explicit methods are applied. In particular, the (linear) diffusive and stabilization terms are discretized implicitly, whereas the non linear convective term is treated by an explicit extrapolation method.
An error analysis of the procedure is carried out, obtaining a priori asymptotic error estimates in the discrete \(L^{\infty}(L^2(\Omega))\)-norm and the \(L^2(H^1(\Omega)\)-seminorm with respect to the mesh size \(h\) and time step \(\tau\) of the form \(\mathcal{O}(h^{p+1} + \tau^k)\) and \(\mathcal{O}(h^{p} + \tau^k)\), respectively, with \(k=2,3\). These theoretical results are illustrated with several numerical examples.

MSC:
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds 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
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
35K55 Nonlinear parabolic equations
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