A stable conservative Lagrange-Galerkin scheme to pure convection equations with mesh intersection

Abstract

We present an algorithm to solve pure-convection problems with a conservative Lagrange-Galerkin formulation in the framework of the finite element method. The weak formulation involves integrals of the product of basis functions associated to the finite element spaces of two different meshes: the original mesh and another one obtained by moving the original mesh along the characteristic curves of the convection operator. In order to compute these integrals up to machine precision, we perform a mesh intersection algorithm which is easily implemented by means of standard finite element operations. Specifically, we consider the intersection of meshes composed by triangles (in 2-dimensions) and tetrahedra (in 3-dimensions) with straight sides. We illustrate the good features of the method in terms of stability, accuracy and mass conservations in different pure-convection tests with velocity fields which are not necessarily divergence-free.