Numerical Analysis of Convection‐Diffusion‐Reaction Problems with Higher Order Characteristics/Finite Elements. Part II: Fully Discretized Scheme and Quadrature Formulas

In this paper a higher order Lagrange–Galerkin discretization method is analyzed when applied to a variable coefficient convection‐(possibly degenerated) diffusion‐reaction equation with mixed Dirichlet–Robin boundary conditions. In a previous paper [A. Bermúdez, M. R. Nogueiras, and C. Vázquez, SIAM J. Numer. Anal., to appear], the proposed second order time discretization scheme has been rigorously introduced for exact and approximated characteristics. Moreover, the $l^{\infty}(L^2)$ stability property and $l^{\infty}(L^2)$ error estimates of order $O(\Delta t^2)$ have been obtained. As a continuation of that work, consistency error estimates of order $O(\Delta t^2 + h^k)$ are obtained for the fully discretized Lagrange–Galerkin scheme. Moreover, adequate quadrature formulas are proposed for the practical implementation of the method with particular finite element spaces. Finally, some numerical tests illustrate the theoretical results and the performance of the combination of second order Lagrange–Galerkin schemes with quadrature formulas.