One level overlapping Schwarz domain decomposition preconditioners can be viewed as a generalization of block Jacobi preconditioning. The effect of the number of blocks and the amount of overlapping between blocks on the convergence rate is well understood. This paper considers the related issue of the effect of the scheme used to partition the matriz into blocks on the convergence rate of the preconditioned iterative method. Numerical results for Laplace and linear elasticity problems in two

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... d three dimensions are presented. The tentative conclusion is that using overlap tends to decrease the differences between the rates of convergence f o r different partitioning schemes. 2 Solving the elliptic problems 2.1 The problems and their discretization Let Q denote the domain imbedded in R2 or E3 and I ' = roUl7l its boundary. The elliptic problems to be solved are either the Poisson problem with Dirichlet boundary conditions on I ' o and homogeneous Neumann boundary conditions on rl or the equations of linear elasticity in the domain SZ with Dirichlet boundary conditions on r. Let u denote the (scalar) solution of the Poisson problem and 2 the (vector)