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Analysis of semi-implicit DGFEM for nonlinear convection-diffusion problems on nonconforming meshes. (English) Zbl 1121.76033
Summary: The paper deals with the numerical analysis of a scalar nonstationary nonlinear convection-diffusion equation. The space discretization is carried out by 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 diffusion and stabilization terms are treated implicitly, whereas nonlinear convective terms are treated explicitly. We derive a priori asymptotic error estimates in the discrete \(L^{\infty}(L^{2})\)-norm, \(L^{2}(H^{1})\)-seminorm and \(L^{\infty}(H^{1})\)-seminorm with respect to the mesh size \(h\) and time step \(\tau\). Numerical examples demonstrate the accuracy of the method and manifest the effect of nonconvexity of elements and nonconformity of the mesh.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76R99 Diffusion and convection
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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