Analysis of semi-implicit DGFEM for nonlinear convection–diffusion problems on nonconforming meshes

The paper deals with the numerical analysis of a scalar nonstationary nonlinear convection–diffusion equation. The space discretization is carried out by the discontinuous Galerkin finite element method (DGFEM), on general nonconforming meshes formed by possibly nonconvex elements, with nonsymmetric treatment of stabilization terms and interior and boundary penalty. The time discretization is carried out by a semi-implicit Euler scheme, in which the diffusion and stabilization terms are treated implicitly, whereas the nonlinear convective terms are treated explicitly. We derive a priori asymptotic error estimates in the discrete L∞(L2)L∞(L2)-norm, L2(H1)L2(H1)-seminorm and L∞(H1)L∞(H1)-seminorm with respect to the mesh size h and time step τ. Numerical examples demonstrate the accuracy of the method and manifest the effect of nonconvexity of elements and nonconformity of the mesh.