This paper describes a parallel iterative solver for finite element discretisations of elliptic partial differential equations on 2D and 3D domains using unstructured grids. The discretisation of the PDE is assumed to be given in the form of element stiffness matrices and the solver is automatic in the sense that it requires minimal additional information about the PDE and the geometry of the domain. The solver parallelises matrix-vector operations required by iterative methods and provides parallel additive Schwarz preconditioners. Parallelisation is implemented through MPI. The paper contains numerical experiments showing almost optimal speedup on unstructured mesh problems on a range of four platforms and in addition gives illustrations of the use of the package to investigate several questions of current interest in the analysis of Schwarz methods. The package is available in public domain from the home page http://www.maths.bath.ac.uk/~mjh/

This paper is concerned with the performance of the conjugate gradient(CG) method with additive Schwarz preconditioner for computing unstructured finite element approximations to the elliptic problem r:aru = f; on\Omega ; u = g on @\Omega D ; @u @n = ~ g on @\Omega N : (1) Here\Omega ae IR 3 is a polyhedral domain with boundary @\Omega partitioned into disjoint subsets @\Omega D 6= ; and @\Omega N , each of which is composed of unions of polygons, and f , g and ~ g are suitably smooth given data. (Analogous results also hold in 2D.) We also assume that a is piecewise constant on each of d open disjoint polyhedral regions k , such that [ d k=1 k = \Omega , and we write aj k = a k where each a k 2 IR + := (0; 1) is constant. We have in mind that the regions k of different material properties are fixed but may have complicated geometry and that the overall mesh used to compute u accurately will be finer than the geometry of the k . There are many applications of this type of problem, for example in groundwater flow and in electromagnetics. After discretisation with linear finite elements on a triangulation T of \Omega\Gamma (1) reduces to the SPD system K(a)x = b(a); (2) where the stiffness matrix and load vector depend continuously on a 2 IR d + . Let h denote the diameter of T and J = max k;l fa k =a l g. It is a standard result that K(a) is ill-conditioned in the sense that (under suitable assumptions) (K(a)) = O(h \Gamma2 ) as h ! 0 for fixed a and (K(a)) = O(J ) as J ! 1 for fixed h. (Here denotes the 2-norm condition number.) One of the striking successes of domain decomposition methods has been the construction of preconditioners for which the condition number of the preconditioned problem is bounded independently of both h and a. ...

This paper is concerned with the performance of the conjugate gradient(CG) method with additive Schwarz preconditioner for computing unstructured finite element approximations to the elliptic problem