First-order System Least- Squares for Second-order Partial Diierential Equations: Part Ii. Siam J

analysis of iterative substructuring algorithms for elliptic problems in three dimensions. Least-squares mixed-nite elements for second-order elliptic problems. A least-squares approach based on a discrete minus one inner product for rst order systems. Technical report, Brookhaven National Laboratory, 1994. 4] J. H. Bramble and J. E. Pasciak. Least-squares methods for Stokes equations based on a discrete minus one inner product.order system least-squares for second-order partial diierential equations: Least-squares mixed nite elements for non-selfadjoint elliptic problems: II. performance of block-ILU factorization methods. Iterative substructuring. Table 6 shows the results for the iterative sub-structuring methods P is with xed subdomain size. They clearly show a constant bound for the condition number and the number of iterations. 5 Conclusions In this paper, some domain decomposition algorithms have been introduced for the discrete systems arising from rst-order system least squares methods applied to second-order elliptic problems. These recently proposed methods allow the use of standard nite element spaces, which are not required to satisfy the inf-sup condition. The analysis of the domain decomposition algorithms follows from analogous results for the standard Galerkin case and the equivalence between the bilinear form associated with the least squares functional and the H 1 (() d+1 norm. Optimal convergence bounds have been proven for overlapping algorithms (additive, multiplicative, coupled, uncoupled versions), while quasi-optimal bounds have been proven for iterative substructuring algorithms. Numerical experiments on a simple model problem connrm these bounds. Future work will investigate the performance of these algorithms for problems with convection and for elliptic systems.