This paper deals with parallelization methods for time-dependent problems where the time steps are shared out among the processors. A Full Multigrid technique serves as solution algorithm, hence information of the preceding time step and of the coarser grid is necessary to compute the solution at each new grid level. Applying the usual extrapolation formula to process this information, the parallelization will not be very efficient. We developed another extrapolation technique which causes a much higher parallelization effect. Test examples show that no essential loss of exactness appears, such that the method presented here shall be well-applicable.

Abstract P-version finite element method for the second order elliptic equation in an arbitrary sufficiently smooth domain is studied in the frame of DD method. Two types square reference elements are used with the products of the integrated Legendre's polynomials for the coordinate functions. There are considered the estimates for the condition numbers, preconditioning of the problems arising on subdomains and the Schur complement, the derivation of the DD preconditioner. For the result we obtain the DD preconditioner to which corresponds the generalized condition number of order (logp )2 . The paper consists of two parts. In part I there are given some preliminary re- sults for 1D case, condition number estimates and some inequalities for 2D reference element.

Introduction In recent years, domain decomposition methods have been used extensively to efficiently solve boundary value problems for partial differential equations in complex{form domains. On the other hand, multilevel techniques on hierarchical data structures also have developed into an effective tool for the construction and analysis of fast solvers. But direct realization of multilevel techniques on a parallel computer system for the global problem in the original domain involves difficult communication problems. I this paper, we present and analyze a combination of these two approaches: domain decomposition and multilevel decomposition on hierarchical structures to design optimal preconditioning operators.

P-version finite element method for the second order elliptic equation in an arbitrary sufficiently smooth domain is studied in the frame of DD method. Two types square reference elements are used with the products of the integrated Legendre's polynomials for the coordinate functions. There are considered the estimates for the condition numbers, preconditioning of the problems arising on subdomains and the Schur complement, the derivation of the DD preconditioner. For the result we obtain the DD preconditioner to which corresponds the generalized condition number of order (logp )2 . The paper consists of two parts. In part I there are given some preliminary results for 1D case, condition number estimates and some inequalities for 2D reference element. Part II is devoted to the derivation of the Schur complement preconditioner and conditionality number estimates for the p-version finite element matrixes. Also DD preconditioning is considered.

This paper deals with parallelization methods for time-dependent problems where the time steps are shared out among the processors. A Full Multigrid technique serves as solution algorithm, hence information of the preceding time step and of the coarser grid is necessary to compute the solution at each new grid level. Applying the usual extrapolation formula to process this information, the parallelization will not be very efficient. We developed another extrapolation technique which causes a much higher parallelization effect. Test examples show that no essential loss of exactness appears, such that the method presented here shall be well-applicable.

In the paper, the parallelization of multi-grid methods for solving second-order elliptic boundary value problems in two-dimensional domains is discussed. The parallelization strategy is based on a non-overlapping domain decomposition data structure such that the algorithm is well-suited for an implementation on a parallel machine with MIMD architecture. For getting an algorithm with a good paral- lel performance it is necessary to have as few communication as possible between the processors. In our implementation, communication is only needed within the smoothing procedures and the coarse-grid solver. The interpolation and restriction procedures can be performed without any communication. New variants of smoothers of Gauss-Seidel type having the same communication cost as Jacobi smoothers are presented. For solving the coarse-grid systems iterative methods are proposed that are applied to the corresponding Schur complement system. Three numerical examples, namely a Poisson equation, a magnetic field problem, and a plane linear elasticity problem, demonstrate the efficiency of the parallel multi- grid algorithm.

Introduction In recent years, domain decomposition methods have been used extensively to efficiently solve boundary value problems for partial differential equations in complex{form domains. On the other hand, multilevel techniques on hierarchical data structures also have developed into an effective tool for the construction and analysis of fast solvers. But direct realization of multilevel techniques on a parallel computer system for the global problem in the original domain involves difficult communication problems. I this paper, we present and analyze a combination of these two approaches: domain decomposition and multilevel decomposition on hierarchical structures to design optimal preconditioning operators.

Abstract P-version finite element method for the second order elliptic equation in an arbitrary sufficiently smooth domain is studied in the frame of DD method. Two types square reference elements are used with the products of the integrated Legendre's polynomials for the coordinate functions. There are considered the estimates for the condition numbers, preconditioning of the problems arising on subdomains and the Schur complement, the derivation of the DD preconditioner. For the result we obtain the DD preconditioner to which corresponds the generalized condition number of order (logp )2 . The paper consists of two parts. In part I there are given some preliminary re- sults for 1D case, condition number estimates and some inequalities for 2D reference element.

P-version finite element method for the second order elliptic equation in an arbitrary sufficiently smooth domain is studied in the frame of DD method. Two types square reference elements are used with the products of the integrated Legendre's polynomials for the coordinate functions. There are considered the estimates for the condition numbers, preconditioning of the problems arising on subdomains and the Schur complement, the derivation of the DD preconditioner. For the result we obtain the DD preconditioner to which corresponds the generalized condition number of order (logp )2 . The paper consists of two parts. In part I there are given some preliminary results for 1D case, condition number estimates and some inequalities for 2D reference element. Part II is devoted to the derivation of the Schur complement preconditioner and conditionality number estimates for the p-version finite element matrixes. Also DD preconditioning is considered.

Extrapolation methods for the solution of partial differential equations are commonly based on the existence of error expansions for the approximate solution. Implicit extrapolation, in the contrast, is based on applying extrapolation indirectly, by using it on quantities like the residual. In the context of multigrid methods, a special technique of this type is known as \034 -extrapolation. For finite element systems this algorithm can be shown to be equivalent to higher order finite elements. The analysis is local and does not use global expansions, so that the implicit extrapolation technique may be used on unstructured meshes and in cases where the solution fails to be globally smooth. Furthermore, the natural multilevel structure can be used to construct efficient multigrid and multilevel preconditioning techniques. The effectivity of the method is demonstrated for heat conduction problems and problems from elasticity theory.