The frequency decomposition multi-grid method

Summary The convergence analysis of a special variant of the additive Schwarz iteration is presented. It can be applied to the frequency decomposition multigrid method and yields robust convergence.     

The frequency decomposition multi-grid method

Summary A new variant of the multi-grid algorithms is presented. It uses multiple coarse-grid corrections with particularly associated prolongations and restrictions. In this paper the robustness with respect to anisotropic problems is considered.Dedicated to the memory of Peter Henrici     

On multi-grid methods for variational inequalities

Summary We consider here a general class of algorithms for the numerical solution of variational inequalities. A convergence proof is given and in particular a multi-grid method is described. Numerical results are presented for the finite-difference discretization of an obstacle problem for minimal surfaces     

Application of domain decomposition methods to isogeometric analysis

During the past decade various new spatial discretization techniques have been developed. In particular, the usage of NURBS based shape functions, well known to the CAD community, has been adapted to finite element technology. In the present work we use the mortar finite element method for the coupling of nonconforming discretized sub-domains in the framework of nonlinear elasticity. We show that the method can be applied to isogeometric analysis with little effort, once the framework of NURBS based shape functions has been implemented. Furthermore, a specific coordinate augmentation technique allows the design of an energy-momentum scheme for the constrained mechanical system under consideration. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

General theory of domain decomposition: Indirect methods

According to a general theory of domain decomposition methods (DDM), recently proposed by Herrera, DDM may be classified into two broad categories: direct and indirect (or Trefftz‐Herrera methods). This article is devoted to formulate systematically indirect methods and apply them to differential equations in several dimensions. They have interest since they subsume some of the best‐known formulations of domain decomposition methods, such as those based on the application of Steklov‐Poincaré operators. Trefftz‐Herrera approach is based on a special kind of Green's formulas applicable to discontinuous functions, and one of their essential features is the use of weighting functions which yield information, about the sought solution, at the internal boundary of the domain decomposition exclusively. A special class of Sobolev spaces is introduced in which boundary value problems with prescribed jumps at the internal boundary are formulated. Green's formulas applicable in such Sobolev spaces, which contain discontinuous functions, are established and from them the general framework for indirect methods is derived. Guidelines for the construction of the special kind of test functions are then supplied and, as an illustration, the method is applied to elliptic problems in several dimensions. A nonstandard method of collocation is derived in this manner, which possesses significant advantages over more standard procedures. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 296–322, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10008     

The multipliers‐free domain decomposition methods

This article concludes the development and summarizes a new approach to dual‐primal domain decomposition methods (DDM), generally referred to as “the multipliers‐free dual‐primal method.” Contrary to standard approaches, these new dual‐primal methods are formulated without recourse to Lagrange‐multipliers. In this manner, simple and unified matrix‐expressions, which include the most important dual‐primal methods that exist at present are obtained, which can be effectively applied to floating subdomains, as well. The derivation of such general matrix‐formulas is independent of the partial differential equations that originate them and of the number of dimensions of the problem. This yields robust and easy‐to‐construct computer codes. In particular, 2D codes can be easily transformed into 3D codes. The systematic use of the average and jump matrices, which are introduced in this approach as generalizations of the “average” and “jump” of a function, can be effectively applied not only at internal‐boundary‐nodes but also at edges and corners. Their use yields significant advantages because of their superior algebraic and computational properties. Furthermore, it is shown that some well‐known difficulties that occur when primal nodes are introduced are efficiently handled by the multipliers‐free dual‐primal method. The concept of the Steklov–Poincaré operator for matrices is revised by our theory and a new version of it, which has clear advantages over standard definitions, is given. Extensive numerical experiments that confirm the efficiency of the multipliers‐free dual‐primal methods are also reported here. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Preconditioning for radial basis functions with domain decomposition methods

In our previous work, an effective preconditioning scheme that is based upon constructing least-squares approximation cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. The preconditioner costs O(N²) flops to set up and O(N) storage. The preconditioning technique is sufficiently general that it can be applied to different types of different operators. This was applied to the 2D multiquadric method, with c~1/√N on the Poisson test problem, the preconditioned GMRES converges in tens of iterations. In this paper, we combine the RBF methods and the ACBF preconditioning technique with the domain decomposition method (DDM). We studied different implementations of the ACBF-DDM scheme and provide numerical results for N > 10,000 nodes. We shall demonstrate that the efficiency of the ACBF-DDM scheme improves dramatically as successively finer partitions of the domain are considered.     

General theory of domain decomposition: Beyond Schwarz methods

Recently, Herrera presented a general theory of domain decomposition methods (DDM). This article is part of a line of research devoted to its further development and applications. According to it, DDM are classified into direct and indirect, which in turn can be subdivided into overlapping and nonoverlapping. Some articles dealing with general aspects of the theory and with indirect (Trefftz–Herrera) methods have been published. In the present article, a very general direct‐overlapping method, which subsumes Schwarz methods, is introduced. Also, this direct‐overlapping method is quite suitable for parallel implementation. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 495–517, 2001     

Superconvergence analysis of two-grid methods for bacteria equations

Numerical Algorithms - In this paper, two-grid methods (TGMs) are developed for a system of reaction-diffusion equations of bacterial infection with initial and boundary conditions. The backward...     

Two-sided approximations for unilateral variational inequalities by multi-grid methods

For the numerical solution of unilateral variational inequalities two iterative schemes are developed which provide approximations from below resp. from above. Both schemes are based on some kind of active set strategy and require the solution of an algebraic system of equations at each iteration step which is done by means of multigrid techniques. Convergence results are established and illustrated by some numerical results for the elastic-plastic torsion problem     

Comparisons between multiplicative and additive Schwarz iterations in domain decomposition methods

We consider the multiplicative and additive Schwarz methods for solving linear systems of equations and we compare their asymptotic rate of convergence. Moreover, we compare the multiplicative Schwarz method with the weighted restricted additive Schwarz method. We prove that the multiplicative Schwarz method is the fastest method among these three. Our comparisons can be done in the case of exact and inexact subspaces solves. In addition, we analyse two ways of adding a coarse grid correction – multiplicatively or additively. Mathematics Subject Classification (1991):65F10, 65F35, 65M55     

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     

New constructions of domain decomposition methods for systems of PDEs

We propose new domain decomposition methods for systems of partial differential equations in two and three dimensions. The algorithms are derived with the help of the Smith factorization. This could also be validated by numerical experiments. To cite this article: V. Dolean et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

A relaxation procedure for domain decomposition methods using finite elements

Summary We present the convergence analysis of a new domain decomposition technique for finite element approximations. This technique was introduced in [11] and is based on an iterative procedure among subdomains in which transmission conditions at interfaces are taken into account partly in one subdomain and partly in its adjacent. No global preconditioner is needed in the practice, but simply single-domain finite element solvers are required. An optimal strategy for an automatic selection of a relaxation parameter to be used at interface subdomains is indicated. Applications are given to both elliptic equations and incompressible Stokes equations.     

Lagrangian formulation of domain decomposition methods: A unified theory

Domain decomposition methods based on one Lagrange multiplier have been shown to be very efficient for solving ill-conditioned problems in parallel. Several variants of these methods have been developed in the last ten years. These variants are based on an augmented Lagrangian formulation involving one or two Lagrange multipliers and on mixed type interface conditions between the sub-domains. In this paper, the Lagrangian formulations of some of these domain decomposition methods are presented both from a continuous and a discrete point of view.     

Convergence of domain decomposition methods via semi-classical calculus semiclassical calculus

We present a model hierarchy of hydrodynamic and quasihydrodynamic equations for plasmas consisting of electrons and ions, and give a rigorous proof of the zero-relaxation-time limits in the hydrodynamic equations. described by the Euler equations coupled with a linear or nonlinear Poisson equation. The proof is based on the high energy estimates for the Euler equations together with compactness arguments.     

Contractivity of domain decomposition splitting methods for nonlinear parabolic problems

This work deals with the efficient numerical solution of nonlinear parabolic problems posed on a two-dimensional domain Ω. We consider a suitable decomposition of domain Ω and we construct a subordinate smooth partition of unity that we use to rewrite the original equation. Then, the combination of standard spatial discretizations with certain splitting time integrators gives rise to unconditionally contractive schemes. The efficiency of the resulting algorithms stems from the fact that the calculations required at each internal stage can be performed in parallel.     

Adaptive wavelet frame domain decomposition methods for nonlinear elliptic equations

In this article, we are concerned with the numerical treatment of nonlinear elliptic boundary value problems. Our method of choice is a domain decomposition strategy. Partially following the lines from (Cohen, Dahmen and deVore, SIAM J Numer Anal 41 (2003), 1785–1823; Kappei, Appl Anal J Sci 90 (2011), 1323–1353; Lui, SIAM J Sci Comput 21 (2000), 1506–1523; Stevenson and Werner, Math Comp 78 (2009), 619–644), we develop an adaptive additive Schwarz method using wavelet frames. We show that the method converges with an asymptotically optimal rate and support our theoretical results with numerical tests in one and two space dimensions. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

On Schwarz's domain decomposition methods for elliptic boundary value problems

Summary. We study the additive and multiplicative Schwarz domain decomposition methods for elliptic boundary value problem of order 2 r based on an appropriate spline space of smoothness . The finite element method reduces an elliptic boundary value problem to a linear system of equations. It is well known that as the number of triangles in the underlying triangulation is increased, which is indispensable for increasing the accuracy of the approximate solution, the size and condition number of the linear system increases. The Schwarz domain decomposition methods will enable us to break the linear system into several linear subsystems of smaller size. We shall show in this paper that the approximate solutions from the multiplicative Schwarz domain decomposition method converge to the exact solution of the linear system geometrically. We also show that the additive Schwarz domain decomposition method yields a preconditioner for the preconditioned conjugate gradient method. We tested these methods for the biharmonic equation with Dirichlet boundary condition over an arbitrary polygonal domain using cubic spline functions over a quadrangulation of the given domain. The computer experiments agree with our theoretical results. Received December 28, 1995 / Revised version received November 17, 1998 / Published online September 24, 1999