Numerical experiments have shown that two-level Schwarz methods often perform very well even if the overlap between neighboring subregions is quite small. This is true to an even greater extent for a related algorithm, due to Barry Smith, where a Schwarz algorithm is applied to the reduced linear system of equations that remains after that the variables interior to the subregions have been eliminated. In this paper, a supporting theory is developed.

. Several domain decomposition methods of Neumann-Neumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite elements. We will only consider problems in three dimensions. Several new variants of the basic algorithm are introduced in a Schwarz method framework that provides tools which have already proven very useful in the design and analysis of other domain decomposition and multi-level methods. The Neumann-Neumann algorithms have several advantages over other domain decomposition methods. The subregions, which define the subproblems, only share the boundary degrees of freedom with their neighbors. The subregions can also be of quite arbitrary shape and many of the major components of the preconditioner can be constructed from subprograms available in standard finite element program libraries. However, in its original form, the algorithm lacks a mechanism for global transportation of informatio...

In recent years, competitive domain-decomposed preconditioned iterative techniques have beendeveloped for nonsymmetric elliptic problems. In these techniques, a large problem is divided into many smaller problems whose requirements for coordination can be controlled to allow e ective solution on parallel machines. Acentral question is how tochoose these small problems and how to arrange the order of their solution. Di erent speci cations of decomposition and solution order lead to a plethora of algorithms possessing complementary advantages and disadvantages. In this report we compare several methods, including the additive Schwarz algorithm, the classical multiplicative Schwarz algorithm, an accelerated multiplicative Schwarz algorithm, the tile algorithm, the CGK algorithm, the CSPD algorithm, and also the popular global ILU-family of preconditioners, on some nonsymmetric or inde nite elliptic model problems discretized by nite di erence methods. The preconditioned problems are solved by the unrestarted GMRES method. A version of the accelerated multiplicative Schwarz method is a consistently good performer.

Iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems. A p-version finite element method based on continuous, piecewise Q p functions is considered for second order elliptic problems in three dimensions; this special method can also be viewed as a conforming spectral element method. An iterative method is designed for which the condition number of the relevant operator grows only in proportion to (1 + log p)². This bound is independent of jumps in the coefficient of the elliptic problem across the interfaces between the subregions. Numerical results are also reported which support the theory.

In the present paper, the solution of nonlinear elliptic boundary value problems (b.v.p.) on parallel machines with Multiple Instruction Multiple Data (MIMD) architecture is discussed. Especially, we consider electro--magnetic field problems the numerical solution of which is based on finite element discretizations and a nested Newton solver. For solving the linear systems of algebraic finite element equations in each Newton step, parallel conjugate gradient methods with a Domain Decomposition preconditioner (DD PCG) as well as parallelized global multigrid methods are applied. The implementation of the whole algorithm, i.e. the mesh generation, the generation of the finite element equations, the nested Newton algorithm, the DD PCG method and the global multigrid method, is based on a non--overlapping DD data structure. The efficiency of the parallel DD PCG methods and the parallelized global multigrid methods, which are embedded in the nested Newton solver, are compared. Fur...

We present a new domain decomposition algorithm for the parallel finite element solution of elliptic partial differential equations. As with most parallel domain decomposition methods each processor is assigned one or more subdomains and an iteration is devised which allows the processors to solve their own subproblem(s) concurrently. The novel feature of this algorithm however is that each of these subproblems is defined over the entire domain --- although the vast majority of the degrees of freedom for each subproblem are associated with a single subdomain (owned by the corresponding processor). This ensures that a global mechanism is contained within each of the subproblems tackled and so no separate coarse grid solve is required in order to achieve rapid convergence of the overall iteration. Furthermore, by following the paradigm introduced in [5], it is demonstrated that this domain decomposition solver may be coupled easily with a conventional mesh refinement code, thus...

The finite element method has grown, in the past 40 years, to be a popular method for the simulation of physical systems in science and engineering. The finite element method is used in a wide array of industries. In fact just about any enterprise that makes a physical product can, and probably does, use finite element technology. The success of the finite element method is due in large part to its ability to allow the use of accurate formulation of partial differential equations (PDEs), on arbitrarily general physical domains with complex boundary conditions. Additionally, the rapid growth in the computational power available in todays computers - for an ever more affordable price - has made finite element technology...

A parallel iterative nonoverlapping domain decomposition method is proposed and analyzed for elliptic problems. Each iteration in this method contains two steps. In the first step, at the interface of two subdomains, one subdomain problem requires that Dirichlet data be passed to it from the previous iteration level, while the other subdomain problem requires Neumann data be passed to it. In the second step, we interchange the types of data passing at the interface of the two subdomains. This domain decomposition method is suitable for parallel processing with coarse granularity. Convergence analysis is demonstrated at the differential level by Hilbert space techniques. Numerical results are provided to confirm the convergence theory. Mathematics subject classification (1991): 65N55, 35J25, 65Y05. 1 Introduction The objective of this paper is to propose and analyze a parallel iterative nonoverlapping domain decomposition method. Each iteration in this method contains two steps. In the...

Domain decomposition algorithms based on the Schwarz framework were originally proposed for the h-version finite element method for elliptic problems. In this thesis, we study some Schwarz algorithms for the p-version finite element method, in which increased accuracy is achieved by increasing the degree p of the elements while the mesh is fixed. These iterative algorithms, often of conjugate gradient type, are both parallel and scalable, and therefore very well suited for massively parallel computing. We consider linear, scalar, self adjoint, second order elliptic problems and quadrilateral elements in the finite element discretization. For a class of overlapping methods, we prove a constant bound, independent of the degree p, the mesh size H and the number of elements N , for the condition number of the iteration operator. This optimal result holds in two and three dimensions for additive and multiplicative schemes, as well as variants on the interface. We consider then local refinem...

: Domain-decomposition methods have been studied and applied to the solution of engineering problems for many years. Recent advances in parallel computing systems have rekindled interest in these methods, at least in part due to the obvious parallelism that is apparent in such algorithms. This paper briefly reviews a number of different domain-decomposition algorithms and shows how they may be used in conjunction with iterative solution techniques for the solution of linear systems of equations arising in the finite element solution of computational mechanics problems. Particular emphasis is placed on so-called iterative substructuring algorithms, in which non-overlapping subdomains are used, and details of how to produce efficient parallel implementations are included. 1 Introduction In this paper we are concerned with the finite element solution of the following second order model problem. Find u 2 V ae H 1 (\Omega\Gamma such that A(u; v) = F (v); 8v 2 V ; (1) where\Omega 2 ! ...