Analysis of a stabilized finite element approximation of the transient convection-diffusion equation using an ale framework

Abstract

In this paper we analyze a stabilized finite element method to solve the transient convection-diffusion-reaction equation based on the decomposition of the unknowns into resolvable and subgrid scales. We start from the time-discrete form of the problem and obtain an evolution equation for both components of the decomposition. A closed-form expression is proposed for the subscales which, when inserted into the equation for the resolvable scale, leads to the stabilized formulation that we analyze. Optimal error estimates in space are provided for the first order, backward Euler time integration.