Overlapping schwarz methods with a standard coarse space for almost incompressible linear elasticity

Low-order finite element discretizations of the linear elasticity system suffer increasingly from locking effects and ill-conditioning, when the material approaches the incompressible limit, if only the displacement variables are used. Mixed finite elements using both displacement and pressure variables provide a well-known remedy, but they yield larger and indefinite discrete systems for which the design of scalable and efficient iterative solvers is challenging. Two-level overlapping Schwarz preconditioners for the almost incompressible system of linear elasticity, discretized by mixed finite elements with discontinuous pressures, are constructed and analyzed. The preconditioned systems are accelerated either by a GMRES (generalized minimum residual) method applied to the resulting discrete saddle point problem or by a PCG (preconditioned conjugate gradient) method applied to a positive definite, although extremely ill-conditioned, reformulation of the problem obtained by eliminating all pressure variables on the element level. A novel theoretical analysis of the algorithm for the positive definite reformulation is given by extending some earlier results by Dohrmann and Widlund. The main result of the paper is a bound on the condition number of the algorithm which is cubic in the relative overlap and grows logarithmically with the number of elements across individual subdomains but is otherwise independent of the number of subdomains, their diameters and mesh sizes, the incompressibility of the material, and possible discontinuities of the material parameters across the subdomain interfaces. Numerical results in the plane confirm the theory and also indicate that an analogous result should hold for the saddle point formulation, as well as for spectral element discretizations

Adaptive selection of primal constraints for isogeometric BDDC deluxe preconditioners

Isogeometric analysis has been introduced as an alternative to finite element methods in order to simplify the integration of computer-aided design (CAD) software and the discretization of variational problems of continuum mechanics. In contrast with the finite element case, the basis functions of isogeometric analysis are often not nodal. As a consequence, there are fat interfaces which can easily lead to an increase in the number of interface variables after a decomposition of the parameter space into subdomains. Building on earlier work on the deluxe version of the BDDC (balancing domain decomposition by constraints) family of domain decomposition algorithms, several adaptive algorithms are developed in this paper for scalar elliptic problems in an effort to decrease the dimension of the global, coarse component of these preconditioners. Numerical experiments provide evidence that this work can be successful, yielding scalable and quasi-optimal adaptive BDDC algorithms for isogeometric discretizations

Isogeometric BDDC Preconditioning with Deluxe Scaling

A balancing domain decomposition by constraints (BDDC) preconditioner with a novel scaling, introduced by Dohrmann for problems with more than one variable coefficient and here denoted as deluxe scaling, is extended to isogeometric analysis of scalar elliptic problems. This new scaling turns out to be more powerful than the standard ?- and stiffness scalings considered in a previous isogeometric BDDC study. Our h-analysis shows that the condition number of the resulting deluxe BDDC preconditioner is scalable with a quasi-optimal polylogarithmic bound which is also independent of coefficient discontinuities across subdomain interfaces. Extensive numerical experiments support the theory and show that the deluxe scaling yields a remarkable improvement over the older scalings, in particular for large isogeometric polynomial degree and high regularity

BDDC and FETI--DP preconditioners for spectral element discretizations of almost incompressible elasticity

We construct and study a BDDC (Balancing Domain Decomposition by Constraints) algorithm, see [1, 2], for the system of almost incompressible elasticity discretized with Gauss Lobatto Legendre (GLL) spectral elements. Related FETIDP algorithms could be considered as well. We show that sets of primal constraints can be found so that these methods have a condition number that depends only weakly on the polynomial degree, while being independent of the number of subdomains (scalability) and of the Poisson ratio and Youngs modulus of the material considered (robustness)

Balancing Neumann-Neumann methods for incompressible Stokes equations

Balancing Neumann-Neumann methods are introduced and studied for incompressible Stokes equations discretized with mixed finite or spectral elements with discontinuous pressures. After decomposing the original domain of the problem into nonoverlapping subdomains, the interior unknowns, which are the interior velocity component and all except the constant-pressure component, of each subdomain problem are implicitly eliminated. The resulting saddle point Schur complement is solved with a Krylov space method with a balancing Neumann-Neumann preconditioner based on the solution of a coarse Stokes problem with a few degrees of freedom per subdomain and on the solution of local Stokes problems with natural and essential boundary conditions on the subdomains. This preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively. The condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and a factor that depends on the inverse of the inf-sup constants of the discrete problem and of the coarse subproblem. Numerical results show that the method is quite fast; they are also fully consistent with the theory

A Hierarchical Preconditioner For The Mortar Finite Element Method

Mortar elements form a family of nonconforming finite element methods that are more flexible than conforming finite elements and are known to be as accurate as their conforming counterparts. A fast iterative method is developed for linear, second order elliptic equations in the plane. Our algorithm is modeled on a hierarchical basis preconditioner previously analyzed and tested, for the conforming case, by Barry Smith and the second author. A complete analysis and results of numerical experiments are given for lower order mortar elements and geometrically conforming decompositions of the region into subregions. Copyright ©1996, Kent State University.47588Achdou, Y., Kuznetsov, Y.A., Substructuring preconditioners for finite element methods on nonmatching grids (1995) East-West J. Numer. Math., 3, pp. 1-28Achdou, Y., Kuznetsov, Y.A., Pironneau, O., Substructuring preconditioners for the Qi mortar element method (1995) Numer. Math., 71, pp. 419-449Achdou, Y., Maday, Y., Widlund, O.B., Methode iterative de sous-structuration pour les elements avec joints (1996) C.R. Acad. Sci. Paris, 322, pp. 185-190Achdou, Y., Maday, Y., Widlund, O.B., Iterative Substructuring preconditioners for the mortar finite element method in two dimensions (1996) Tech. Rep., Courant Institute of Mathematical Sciences, , In preparationBen Belgacem, F., Discretisations 3D Non Conformes pour la Methode de Decomposition de Domaine des Element avec Joints: Analyse Mathematique et Mise en vre pour le Probleme de Poisson (1993) Tech. Rep. HI-72/93017, , PhD thesis, Universite Pierre et Marie Curie, Paris, France, January Electricite de FranceBen Belgacem, F., Maday, Y., Adaption de la methode des elements avec joints au couplage spectral elments finis en dimension 3: Etude de l'erreur pour l'equation de Poisson, tech. rep., Electricite de France, April 1992 Tech. Rep. HI-72/7095.The mortar element method for three dimensional finite elements (1993) Unpublished Paper Based on Yvon Maday's Talk at the Seventh International Conference of Domain Decomposition Methods in Scientific and Engineering Computing, Held at Penn State University, , October 27-30Bernardi, C., Maday, Y., Mesh adaptivity in finite elements by the mortar method (1995) Tech. Rep. R94029, , Laboratoire d'Analyse Numerique, Universite Pierre et Marie Curie -Centre National de la Recherche Scientifique, JanuaryBernardi, C., Maday, Y., Patera, A.T., A new non conforming approach to domain decomposition: The mortar element method (1994) Collège De France Seminar, , H. Brezis and J.-L. Lions, eds., PitmanBjrstad, P.E., Wldlund, O.B., Iterative methods for the solution of elliptic problems on regions partitioned into substructures (1986) SIAM J. Numer. Anal., 23, pp. 1093-1120Bramble, J.H., A second order finite difference analogue of the first biharmonic boundary value problem (1966) Numer. Math., 9, pp. 236-249Casarin, M.A., Diagonal edge preconditioners in p-version and spectral element methods (1995) Tech. Rep., 704. , Department of Computer Science, Courant Institute, September SIAM J. Sci. ComputClarlet, P.G., (1978) The Finite Element Method for Elliptic Problems, , North-Holland, AmsterdamDryja, M., Additive Schwarz methods for elliptic mortar finite element problems (1996) Modeling and Optimization of Distributed Parameter Systems with Applications to Engineering, , K. Malanowski, Z. Nahorski, and M. Peszynska, eds., IFIP, Chapman & Hall, London, To appearDryja, M., Wldlund, O.B., Schwarz methods of Neumann-Neumann type for threedimensional elliptic finite element problems (1995) Comm. Pure Appl. Math., 48, p. 121155Tallec, P.L.E., Neumann-Neumann domain decomposition algorithms for solving 2D elliptic problems with nonmatching grids (1993) East-West J. Numer. Math., 1, pp. 129-146Maday, Y., Wldlund, O.B., Some iterative sub structuring methods for mortar finite elements: The lower order case (1996) Tech. Rep., Courant Institute of Mathematical Sciences, , In preparationSmith, B.F., Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity Tech. Rep., 517. , PhD thesis, Courant Institute of Mathematical Sciences, September 1990. Department of Computer Science, Courant InstituteSmith, B.F., Wldlund, O.B., A domain decomposition algorithm using a hierarchical basis (1990) SIAM J. Sci. Stat. Comput., 11, pp. 1212-1220Wldlund, O.B., Iterative sub structuring methods: Algorithms and theory for elliptic problems in the plane (1988) First International Symposium on Domain Decomposition Methods for Partial Differential Equations, , R. Glowinski, G. H. Golub, G. A. Meurant, and J. Periaux, eds., Philadelphia, PA, SIAMYserentant, H., On the multi-level splitting of finite element spaces (1986) Numer. Math., 49, pp. 379-41