Analysis of a BDF–DGFE scheme for nonlinear convection–diffusion problems

• Published in: Numerische Mathematik 2008-10, Vol.110 (4), p.405-447
• Main Authors: DOLEJSI, Vit, VLASAK, Miloslav
• Format: Article
• Language: English
• Subjects: Acceleration of convergence; Calculus of variations and optimal control; Convection-diffusion equation; Discretization; Exact sciences and technology; Finite element method; Linear algebra; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical analysis. Scientific computation; Numerical and Computational Physics; Ordinary differential equations; Sciences and techniques of general use; Simulation; Theoretical
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-008-0178-2

We deal with the numerical solution of a scalar nonstationary nonlinear convection–diffusion equation. We employ a combination of the discontinuous Galerkin finite element method for the space semi-discretization and the k -step backward difference formula for the time discretization. The diffusive and stabilization terms are treated implicitly whereas the nonlinear convective term is treated by a higher order explicit extrapolation method, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the discrete L ∞ ( L 2 )-norm and the L 2 ( H 1 )-seminorm with respect to the mesh size h and time step τ for k  = 2,3. Numerical examples verifying the theoretical results are presented.