INCREMENTAL UNKNOWNS FOR CONVECTION DIFFUSION-EQUATIONS

When iterative methods (e.g. MR, GCR, Orthomin(k), see [3]) are used to approximate the solution of a nonsymmetric linear system AU = b, where A is an N x N matrix with positive symmetric part, i.e. M = (A + A(t))/2 is positive-definite, the convergence rates depend on the number nu(A) = lambda(max)(A(t)A)1/2/lambda(min)(M) which is the condition number of A when A is symmetric positive-definite. In this paper, we use incremental unknowns (IU) see [1,2,5]) in conjunction with the above iterative methods to approximate the solution of the nonsymmetric linear system generated from the discretization of a convection-diffusion equation. We show theoretically that, with the use of IU, nu is of order O((log(h))2) instead of O(1/h2) which is the order of nu with the use of the usual nodal unknowns, where h is the mesh size for the finite differences. With the use of (2.3) in Theorem 2.1, we can show that at most O((log(h))4) iterations are needed to attain an acceptable solution. Numerical results are also presented. They show that actually, O(\log(h)\) iterations are needed with the use of IU and O(1/h) iterations are needed with the use of nodal unknowns to obtain the approximate solution. The algorithms are efficient and easy to implement.