Numerical Analysis of a Second order Pure Lagrange–Galerkin Method for Convection-Diffusion Problems. Part I: Time Discretization

Abstract

[Abstract]: We propose and analyze a second order pure Lagrangian method for variable coefficient convection-(possibly degenerate) diffusion equations with mixed Dirichlet-Robin boundary conditions. First, the method is rigorously introduced for exact and approximate characteristics. Next, l1(H1) stability is proved and l1(H1) error estimates of order O(Δt2) are obtained. Moreover, l1(L2) stability and l1(L2) error estimates of order O(Δt2) with constants bounded in the hyperbolic limit are shown. For the particular case of Dirichlet boundary conditions, diffusion tensor A = ϵI and right-hand side f = 0, the l1(H1) stability estimate is independent of ϵ. Moreover, for incompressible flows the constants in the stability inequalities are independent of the final time. In a second part of this work, the pure Lagrangian scheme will be combined with Galerkin discretization using finite elements spaces and numerical examples will be presented.