Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements

Summary. Two-level domain decomposition methods are developed for a simple nonconforming approximation of second order elliptic problems. A bound is established for the condition number of these iterative methods, that grows only logarithmically with the number of degrees of freedom in each subregion. This bound holds for two and three dimensions and is independent of jumps in the value of the coefficients and number of subregions. We introduce face coarse spaces, and isomorphisms to map between conforming and nonconforming spaces. ReceivedMarch 1, 1995 / Revised version received January 16, 1996     

Analysis of two new parareal algorithms based on the Dirichlet-Neumann/Neumann-Neumann waveform relaxation method for the heat equation

The Dirichlet-Neumann and Neumann-Neumann waveform relaxation methods are nonoverlapping spatial domain decomposition methods to solve evolution problems, while the parareal algorithm is in time parallel fashion. Based on the combinations of these space and time parallel strategies, we present and analyze two parareal algorithms based on the Dirichlet-Neumann and the Neumann-Neumann waveform relaxation method for the heat equation by choosing Dirichlet-Neumann/Neumann-Neumann waveform relaxation as two new kinds of fine propagators instead of the classical fine propagator. Both new proposed algorithms could be viewed as a space-time parallel algorithm, which increases the parallelism both in space and in time. We derive for the heat equation the convergence results for both algorithms in one spatial dimension. We also illustrate our theoretical results with numerical experiments finally.

     

A domain decomposition algorithm for elliptic problems in three dimensions

Summary Most domain decomposition algorithms have been developed for problems in two dimensions. One reason for this is the difficulty in devising a satisfactory, easy-to-implement, robust method of providing global communication of information for problems in three dimensions. Several methods that work well in two dimension do not perform satisfactorily in three dimensions.A new iterative substructuring algorithm for three dimensions is proposed. It is shown that the condition number of the resulting preconditioned problem is bounded independently of the number of subdomains and that the growth is quadratic in the logarithm of the number of degrees of freedom associated with a subdomain. The condition number is also bounded independently of the jumps in the coefficients of the differential equation between subdomains. The new algorithm also has more potential parallelism than the iterative substructuring methods previously proposed for problems in three dimensions.This work was supported in part by the National Science Foundation under grant NSF-CCR-8903003 and by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Balancing Neumann-Neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity

Summary. Balancing Neumann-Neumann methods are extented to mixed formulations of the linear elasticity system with discontinuous coefficients, discretized with mixed finite or spectral elements with discontinuous pressures. These domain decomposition methods implicitly eliminate the degrees of freedom associated with the interior of each subdomain and solve iteratively the resulting saddle point Schur complement using a hybrid preconditioner based on a coarse mixed elasticity problem and local mixed elasticity problems with natural and essential boundary conditions. A polylogarithmic bound in the local number of degrees of freedom is proven for the condition number of the preconditioned operator in the constant coefficient case. Parallel and serial numerical experiments confirm the theoretical results, indicate that they still hold for systems with discontinuous coefficients, and show that our algorithm is scalable, parallel, and robust with respect to material heterogeneities. The results on heterogeneous general problems are also supported in part by our theory. Mathematics Subject Classification (1991):65N55, 65N30, 65N35, 65F10, 65Y05This work was supported by a scholarship of CNPq, of the Ministry for Science and Technology of Brazil, under process 201205/97-1. The work was developed in part at MCS/ANL-DOE, under a Givens Research Associate appointment in Summer 2001.This work was supported in part by the National Science Foundation under Grant NSF-CCR-9732208 and in part by MIUR.This work was supported in part by the National Science Foundation under Grants qNSF-CCR-9732208, and in part by the U.S. Department of Energy under contracts DE-FC02-01ER25482 and DE-FG02-92ER25127.     

A parallel algorithm for partially separable non-convex global minimization: Linear constraints

The global minimization of large-scale partially separable non-convex problems over a bounded polyhedral set using a parallel branch and bound approach is considered. The objective function consists of a separable concave part, an unseparated convex part, and a strictly linear part, which are all coupled by the linear constraints. These large-scale problems are characterized by having the number of linear variables much greater than the number of nonlinear variables. An important special class of problems which can be reduced to this form are the synomial global minimization problems. Such problems often arise in engineering design, and previous computational methods for such problems have been limited to the convex posynomial case. In the current work, a convex underestimating function to the objective function is easily constructed and minimized over the feasible domain to get both upper and lower bounds on the global minimum function value. At each minor iteration of the algorithm, the feasible domain is divided into subregions and convex underestimating problems over each subregion are solved in parallel. Branch and bound techniques can then be used to eliminate parts of the feasible domain from consideration and improve the upper and lower bounds. It is shown that the algorithm guarantees that a solution is obtained to within any specified tolerance in a finite number of steps. Computational results obtained on the four processor Cray 2, both sequentially and in parallel on all four processors, are also presented.     

Overlapping Domain Decomposition Algorithms for General Sparse Matrices

Domain decomposition methods for finite element problems using a partition based on the underlying finite element mesh have been extensively studied. In this paper, we discuss algebraic extensions of the class of overlapping domain decomposition algorithms for general sparse matrices. The subproblems are created with an overlapping partition of the graph corresponding to the sparsity structure of the matrix. These algebraic domain decomposition methods are especially useful for unstructured mesh problems. We also discuss some difficulties encountered in the algebraic extension, particularly the issues related to the coarse solver.     

A hybrid approach to 3-level FETI

Finite Element Tearing and Interconnecting (FETI) methods are nonoverlapping domain decomposition methods which have been proven to be very robust and parallel scalable for a class of elliptic partial differential equations. These methods are also called dual domain decomposition methods since the continuity accross the subdomain boundaries is enforced by Lagrange multipliers and, after elimination of the primal variables, the remaining Schur complement system is solved iteratively in the Lagrange multiplier space using a Krylov space method. Domain decomposition methods iterating on the primal variables are called primal substructuring methods. FETI and FETI–DP methods are different members of the family of dual domain decomposition methods. Their standard versions have in common that the local subproblems and a small global problem are solved exactly by a direct method, essentially representing two different levels within the algorithm. Several extensions of dual and primal iterative substructuring beyond two levels have been proposed in the past, see, e.g., [7] for FETI–DP, and, e.g., Tu [13,12,11] or [9] and [1] for BDDC. In the present article, a hybrid FETI/FETI–DP method is considered and some numerical results are presented. It is noted that independently, there is ongoing research on hybrid FETI methods by Jungho Lee of the Courant Institute. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A domain decomposition method for linear exterior boundary value problems

In this paper, we present a domain decomposition method, based on the general theory of Steklov-Poincaré operators, for a class of linear exterior boundary value problems arising in potential theory and heat conductivity. We first use a Dirichlet-to-Neumann mapping, derived from boundary integral equation methods, to transform the exterior problem into an equivalent mixed boundary value problem on a bounded domain. This domain is decomposed into a finite number of annular subregions, and the Dirichlet data on the interfaces is introduced as the unknown of the associated Steklov-Poincaré problem. This problem is solved with the Richardson method by introducing a Dirichlet-Robin-type preconditioner, which yields an iteration-by-subdomains algorithm well suited for parallel computations. The corresponding analysis for the finite element approximations and some numerical experiments are also provided.     

Path-following gradient-based decomposition algorithms for separable convex optimization

A new decomposition optimization algorithm, called path-following gradient-based decomposition, is proposed to solve separable convex optimization problems. Unlike path-following Newton methods considered in the literature, this algorithm does not require any smoothness assumption on the objective function. This allows us to handle more general classes of problems arising in many real applications than in the path-following Newton methods. The new algorithm is a combination of three techniques, namely smoothing, Lagrangian decomposition and path-following gradient framework. The algorithm decomposes the original problem into smaller subproblems by using dual decomposition and smoothing via self-concordant barriers, updates the dual variables using a path-following gradient method and allows one to solve the subproblems in parallel. Moreover, compared to augmented Lagrangian approaches, our algorithmic parameters are updated automatically without any tuning strategy. We prove the global convergence of the new algorithm and analyze its convergence rate. Then, we modify the proposed algorithm by applying Nesterov’s accelerating scheme to get a new variant which has a better convergence rate than the first algorithm. Finally, we present preliminary numerical tests that confirm the theoretical development.     

A parallel algorithm for constrained concave quadratic global minimization

The global minimization of large-scale concave quadratic problems over a bounded polyhedral set using a parallel branch and bound approach is considered. The objective function consists of both a concave part (nonlinear variables) and a strictly linear part, which are coupled by the linear constraints. These large-scale problems are characterized by having the number of linear variables much greater than the number of nonlinear variables. A linear underestimating function to the concave part of the objective is easily constructed and minimized over the feasible domain to get both upper and lower bounds on the global minimum function value. At each minor iteration of the algorithm, the feasible domain is divided into subregions and linear underestimating problems over each subregion are solved in parallel. Branch and bound techniques can then be used to eliminate parts of the feasible domain from consideration and improve the upper and lower bounds. It is shown that the algorithm guarantees that a solution is obtained to within any specified tolerance in a finite number of steps. Computational results are presented for problems with 25 and 50 nonlinear variables and up to 400 linear variables. These results were obtained on a four processor CRAY2 using both sequential and parallel implementations of the algorithm. The average parallel solution time was approximately 15 seconds for problems with 400 linear variables and a relative tolerance of 0.001. For a relative tolerance of 0.1, the average computation time appears to increase only linearly with the number of linear variables.     

A domain decomposition method for two-body contact problems with Tresca friction

The paper analyzes a continuous and discrete version of the Neumann-Neumann domain decomposition algorithm for two-body contact problems with Tresca friction. Each iterative step consists of a linear elasticity problem for one body with displacements prescribed on a contact part of the boundary and a contact problem with Tresca friction for the second body. To ensure continuity of contact stresses, two auxiliary Neumann problems in each domain are solved. Numerical experiments illustrate the performace of the proposed approach.     

Unsteady convection and convection-diffusion problems via direct overlapping domain decomposition methods

In solving unsteady problems,domain decomposition methods may be used either for iterative preconditioning each global implicit time-step or directly (noniteratively) within a blockwise implicit time-stepping procedure, in the latter case, the inner boundary values for the subproblems are generated by explicit time-extrapolation. The overlapping variants of this method have been proved to be efficient tools for solving parabolic and first-order hyperbolic problems on modern parallel computers, because they require global communication only once per time-step. The mechanism making this possible is the exponential decay in space of the time-discrete Green's function. We investigate several model problems of convection and convection-diffusion. Favorable optimal and far-reaching estimates of the overlap required have been established in the case of exemplary standard upwind finite-difference schemes. In particular, it has been shown that the overlap for the convection-diffusion problem is the additive function of overlaps for the corresponding convection and diffusion problem to be considered independently. These results have been confirmed with several numerical test examples. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 387–406, 1998     

定常Stokes方程一种基于完全区域分解的有限元并行算法

基于完全区域分解技巧,提出了一种求解定常Stokes方程的有限元并行算法．该算法中,所有子问题都是定义在整个求解区域上,但绝大部分自由度来自其所负责的子区域,从而使得算法稍加修改现有的串行程序即可实现相应的并行计算,实现简单,通信需求少．数值结果验证了算法的高效性．     

Two-level Schwarz method for unilateral variational inequalities

The numerical solution of variational inequalities of obstacletype associated with second-order elliptic operators is considered.Iterative methods based on the domain decomposition approachare proposed for discrete obstacle problems arising from thecontinuous, piecewise linear finite element approximation ofthe differential problem. A new variant of the Schwarz methodology,called the two-level Schwarz method, is developed offering thepossibility of making use of fast linear solvers (e.g., linearmultigrid and fictitious domain methods) for the genuinely nonlinearobstacle problems. Namely, by using particular monotonicityresults, the computational domain can be partitioned into (mesh)subdomains with linear and nonlinear (obstacle-type) subproblems.By taking advantage of this domain decomposition and fast linearsolvers, efficient implementation algorithms for large-scalediscrete obstacle problems can be developed. The last part ofthe paper is devoted to illustrate numerical experiments.     

A convergent decomposition algorithm for support vector machines

In this work we consider nonlinear minimization problems with a single linear equality constraint and box constraints. In particular we are interested in solving problems where the number of variables is so huge that traditional optimization methods cannot be directly applied. Many interesting real world problems lead to the solution of large scale constrained problems with this structure. For example, the special subclass of problems with convex quadratic objective function plays a fundamental role in the training of Support Vector Machine, which is a technique for machine learning problems. For this particular subclass of convex quadratic problem, some convergent decomposition methods, based on the solution of a sequence of smaller subproblems, have been proposed. In this paper we define a new globally convergent decomposition algorithm that differs from the previous methods in the rule for the choice of the subproblem variables and in the presence of a proximal point modification in the objective function of the subproblems. In particular, the new rule for sequentially selecting the subproblems appears to be suited to tackle large scale problems, while the introduction of the proximal point term allows us to ensure the global convergence of the algorithm for the general case of nonconvex objective function. Furthermore, we report some preliminary numerical results on support vector classification problems with up to 100 thousands variables.     

Nonoverlapping domain decomposition and additive splitting up methods for multidimensional parabolic problem

The explicit implicit domain decomposition methods are noniterative types of methods for nonoverlapping domain decomposition but due to the use of the explicit step for the interface prediction, the methods suffer from inaccuracy of the usual explicit scheme. In this article a specific type of first‐ and second‐order splitting up method, of additive type, for the dependent variables is initially considered to solve the two‐ or three‐dimensional parabolic problem over nonoverlapping subdomains. We have also considered the parallel explicit splitting up algorithm to define (predict) the interface boundary conditions with respect to each spatial variable and for each nonoverlapping subdomains. The parallel second‐order splitting up algorithm is then considered to solve the subproblems defined over each subdomain; the correction step will then be considered for the predicted interface nodal points using the most recent solution values over the subdomains. Finally several model problems will be considered to test the efficiency of the presented algorithm. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell's Equations

We consider the convergence theory of adaptive multigrid methods for second-order elliptic problems and Maxwell's equations. The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their "immediate" neighbors. In the context of lowest order conforming finite element approximations, we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms. The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures. The convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of freedom. We demonstrate our convergence theory by two numerical experiments.     

A Neumann-Neumann algorithm for a mortar discretization of elliptic problems with discontinuous coefficients

Summary. Neumann-Neumann algorithm have been well developed for standard finite element discretization of elliptic problems with discontinuous coefficients. In this paper, an algorithm of this kind is designed and analyzed for a mortar finite element discretization of problems in three dimensions. It is established that its rate of convergence is independent of the discretization parameters and jumps of coefficients between subregions. The algorithm is well suited for parallel computations.Mathematics Subject Classification (1991): 65N55, 65N10, 65N30, 65N22.The work was supported in part by the U.S. Department of Energy under contract DE-FG02-92ER25127 and in part by Polish Science Foundation under grant 2P03A00524.AcknowledgmentThe author would like to thank Olof Widlund for many fruitful discussions and valuable remarks and suggestions on how to improve the presentation of our results.     

The use of splitting of the equations of motion in the numerical solution of problems of the dynamics of elastic systems

Linear elastic systems with a finite number of degrees of freedom, the initial equations of motion of which are constructed using the finite element method or other discretization methods, are considered. Since, in applied dynamics problems, the motions are usually investigated in a frequency range with an upper bound, the degrees of freedom of the initial system of equations are split into dynamic and quasi-dynamic degrees. Finally, the initial system of equations is split into a small number of differential equations for the dynamic degrees of freedom and into a system of algebraic equations for determining the quasi-static displacements, represented in the form of a matrix series. The number of terms of the series taken into account depends on the accuracy required.     

Coercive domain decomposition algorithms for advection-diffusion equations and systems

Two families of non-overlapping coercive domain decomposition methods are proposed for the numerical approximation of advection-dominated advection-diffusion equations and systems. Convergence is proven for both the continuous and the discrete problem. The rate of convergence of the first method is shown to be independent of the total number of degrees of freedom. Several numerical results are presented, showing the efficiency and robustness of the proposed interative algorithms.