Numerical Analysis of a Second Order Pure Lagrange--Galerkin Method for Convection-Diffusion Problems. Part II: Fully Discretized Scheme and Numerical Results

Abstract

[Abstract]: We analyze a second order pure Lagrange-Galerkin method for variable coefficient convection-(possibly degenerate) diffusion equations with mixed Dirichlet-Robin boundary conditions. In a previous paper the proposed second order pure Lagrangian time discretization scheme has been introduced and analyzed for the same problem. More precisely, the l1(H1) stability and l1(H1) error estimates of order O(_t2) has been obtained. Moreover, for the particular case of incompressible flows, stability inequalities with constants independent of the final time have been stated. In the present paper l1(H1) error estimates of order O(_t2) + O(hk) are obtained for the fully discretized pure Lagrange-Galerkin method. To prove these results we use some properties obtained in the previous paper. Finally, numerical tests are presented that confirm the theoretical results.