A GALERKIN CHARACTERISTIC ALGORITHM FOR TRANSPORT-DIFFUSION EQUATIONS

We introduce in this paper an algorithm for integration of the transpo rt-diffusion equations by combining the Galerkin finite element method with the discretization of the total derivative along the characteris tics. The present algorithm is an extension of the one proposed by the author [R. Bermejo, Numer. Math., 60 (1991), pp. 163-194] to integrat e the transport equation. The algorithm consists of a convective-diffu sive splitting of the equations along the characteristics. Such a sche me permits an efficient evaluation of inner products of functions that take their values in different partitions (grids). The convective sta ge of the splitting is evaluated in a Q(1)-conforming finite element s pace, while the diffusive stage can be approximated in higher-order (Q (2) or Q(3)) finite element spaces. We analyze two time-discretization schemes for the diffusive stage, namely, the backward Euler and Crank -Nicolson schemes. Our L(2)-norm estimate shows that, provided h = O(D elta t), the backward Euler scheme combined with a Q(1)-conforming fin ite element yields optimal error estimates, whereas the Crank-Nicolson scheme is optimal when combined with a Q(2)-conforming finite element .