Pure Lagrangian and Semi-Lagrangian Finite Element Methods for the Numerical Solution of Convection-Diffusion Problems

Abstract

In this paper we propose a unified formulation to introduce and analyze (pure) Lagrangian and semi-Lagrangian methods for solving convection-diffusion partial differential equations. This formulation allows us to state classical and new numerical methods. Several examples are given. We combine them with finite element methods for spatial discretization. One of the pure Lagrangian methods we introduce has been analyzed in [4] and [5] where stability and error estimates for time semi-discretized and fully-discretized schemes have been proved. In this paper, we prove new stability estimates. More precisely, we obtain an $l^∞(H^1)$ stability estimate independent of the diffusion coefficient and, if the underlying flow is incompressible, we get a stability inequality independent of the final time. Finally, the numerical solution of a test problem is presented that confirms the new stability results.