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nonlinear convection-diffusion problem; barycentric finite volumes; Crouzeix-Raviart nonconforming piecewise linear finite elements; monotone finite volume scheme; discrete maximum principle; a priori estimates; convergence of the method
We present the convergence analysis of an efficient numerical method for the solution of an initial-boundary value problem for a scalar nonlinear conservation law equation with a diffusion term. Nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume barycentric mesh, whereas the diffusion term is discretized by piecewise linear nonconforming triangular finite elements. Under the assumption that the triangulations are of weakly acute type, with the aid of the discrete maximum principle, a priori estimates and some compactness arguments based on the use of the Fourier transform with respect to time, the convergence of the approximate solutions to the exact solution is proved, provided the mesh size tends to zero.
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