A PARALLEL ITERATION METHOD AND THE CONVECTION-DIFFUSION EQUATION

A pair of parallel sequences {u(k)} --> u, {v(k)} --> 0 (1.3) is generated to solve the n x n linear system Au = (I(n) - B)u = f. Convergence depends only on the geometry or shape of sigma(B), the set of eigenvalues of B. The parallel method is applied to the singularly perturbed convection-diffusion equation (6.1), when the Reynolds number in the direction of flow is large. Numerical comparisons with known results are given. Our theory also applies to the class of possibly nonsymmetric A with real spectrum (cf. Theorem 5.1) and to several other classes of systems as well. Computations to generate the sequences are relatively straightforwaxd, as is indicated in our main result, Theorem 4.1. In fact, the parameters of the embracing ellipse for sigma(B)2 (4.6) completely determine (i) the coefficients for the parallel sequences {u(k)} --> u and {v(k)} --> 0 and (ii) the spectral radius (4.4), which characterizes their asymptotic convergence rate (2.4). Figure 5.1 illustrates some geometries for sigma(B) that are accommodated by our theory and Figure 7.1 shows the eigenvalue bowtie region arising from the convection-diffusion equation with large Reynolds number.