DOCSLIB.ORG

Adaptive Multilevel BDDC

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Adaptive Multilevel BDDC Adaptive Multilevel BDDC Jan Mandel Includes joint work with Clark Dohrmann, BedˇrichSoused´ık,Jakub S´ıstek,Pavelˇ Burda, Marta Cert´ıkov´a,andˇ Jaroslav Novotn´y. Center for Computational Mathematics and Department of Mathematical and Statistical Sciences University of Colorado Denver Supported by National Science Foundation under grant DMS-0713876 DD19, August 2009 (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 1 / 49 Outline Some (biased) history BDDC formulation Adaptive Multilevel BDDC Implementation by global matrices and generalized change of variables Implementation on top of frontal solver (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 2 / 49 Motivation BDDC{BalancingDomainDecomposition byConstraints From Dohrmann (2003): Some remaining issues need to be addressed for improvement. First, it would be useful to have an effective method for select- ing additional corners and edges to improve performance for very poorly conditioned problems. Second, the performance of the multilevel extension should be investigated further. Recall that the multilevel extension is obtained by recursive application of the preconditioner to coarse problem stiffness matrices. Such an ex- tension would be beneficial for problems with very large numbers of substructures... (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 3 / 49 Basic substructuring assemble elements into substructures, eliminate interiors =) reduced problem (Schur complement) on interface (Dirichlet-to-Neumann operator H1=2 ! H−1=2, a.k.a. Poincar´e-Steklovoperator) for parallel solution; Schur complement matrix-vector multiply = solve substructure Dirichlet problem only matrix datastructures needed; condition number O(1=h) better in practice than the O 1=h2 for the original problem (for small N) (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 4 / 49 Early Preconditioners diagonal preconditioning: Gropp and Keyes 1987, \probing" the diagonal of the Schur complement Chan and Mathew 1992 (because creating the diagonal of the Schur complement is expensive)... preconditioning by solving substructure Neumann problems (H−1=2 ! H1=2) Glowinski and Wheeler 1988, Le Talleck and De Roeck 1991 (a.k.a. the Neumann-Neumann method) add coarse space { asymptotically optimal preconditioners: H=h ! log2 (1 + H=h) (Bramble, Pasciak, Schatz 1986+, Widlund 1987, Dryja 1988, Dryja-Widlund-Smith 1994... ) but all these methods require access to mesh details and depend on details of the Finite Element code, which makes them hard to interface with existing FE software (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 5 / 49 FETI and BDD algebraic - need only substructure 1. solvers (Neumann, Dirichlet), 2. connectivity, 3. basis of nullspace (BDD - constant function for the Laplacian, FETI - actual nullspace) both involve singular substructure problems (Neumann) and build the coarse problem from local substructure nullspaces (in different ways) to assure that the singular systems are consistent Balancing Domain Decomposition (BDD, Mandel 1993): solve the system reduced to interfaces, interface degrees of freedom common (this is the Neumann-Neumann with a particular coarse space and multiplicative coarse correction) Finite Element Tearing and Interconnecting (FETI, Farhat and Roux 1991): enforce continuity across interfaces by Lagrange multipliers, solve the dual system for the multipliers (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 6 / 49 FETI and BDD Developments Both require only matrix level information available in FE software. So they became very popular and widely used. The methods work well in 2D and 3D (solids). But the performance for plates/shells/biharmonic was not so good. Reason: the condition numbers depend on the energy (trace norm) of functions with jumps across a substructure corner. In 2D, OK for H1=2 traces of H1 functions, not H3=2 traces of H2 functions (embedding theorem). Fix: avoid this by increasing the coarse space and so restricting the space where the method runs, to make sure that nothing gets torn across the corners (BDD: LeTallec Mandel Vidrascu 1998, FETI: Farhat Mandel 1998, Farhat Mandel Tezaur 1998). Drawback: complicated, expensive, a large coarse problem with custom basis functions (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 7 / 49 The best so far: FETI-DP and BDDC To assure that nothing gets torn across the corners, enforce identical values at corners from all neighboring substructures a-priori =) corner values are coarse degrees of freedom Continuity elsewhere at the interfaces by Lagrange multipliers as in FETI =) FETI-DP (Farhat et al 2001) Continuity elsewhere by common values as in BDD =) BDDC (Dohrmann 2003; independently Cros 2003, Frakagis and Papadrakakis 2003, with corner coarse degrees of freedom only) Additional coarse degrees of freedom (side/face averages) required in 3D for good conditioning: Farhat, Lesoinne, Pierson 2000 (algorithm only), Dryja Windlud 2002 (with proofs) (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 8 / 49 Evolution of BDD/BDDC/FETI-DP (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 9 / 49 Substructuring for the two-level method (with H/h=4) (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 10 / 49 BDDC description - example of spaces N N W = Wi : space of block vectors, one block per substructure i=1 U ⊂ Wf ⊂ W continuous across whole continuous across no continuity substructure interfaces corners only required global matrix: assembled partially assembled not assembled substructures: assembled assembled assembled Want to solve u 2 U : a (u; v) = hf ; vi 8v 2 U: (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 11 / 49 Abstract two-level BDDC: Variational setting of the problem and algorithm components u 2 U : a(u; v) = hf ; vi ; 8v 2 U form a (·; ·) is SPD on U and positive semidefinite on W ⊃ U, Example: W = W1 × · · · × WN (spaces on substructures) U = functions continuous across interfaces Choose preconditioner components: 1 space Wf, U ⊂ Wf ⊂ W ; such that a (·; ·) is positive definite on Wf . Example: functions with continuous coarse dofs, such as values at substructure corners 2 projection E : Wf ! U; range E = U . Example: averaging across substructure interfaces (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 12 / 49 Abstract BDDC preconditioner Theorem The abstract BDDC preconditioner M : U −! U defined by M : r 7−! u = Ew; w 2 Wf : a (w; z) = hr; Ezi ; 8z 2 Wf: satisfies λ (MA) k(I − E)wk2 κ = max ≤ ! = sup a : λ (MA) 2 min w2Wf kwka The space Wf is defined using so called coarse degrees of freedom, as Wf = fw 2 W : C (I − E) w = 0g ; C ::: weights on the local coarse degrees of freedom, E ::: averaging, =) coarse degrees of freedom on adjacent substructures coincide. (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 13 / 49 Coarse degrees of freedom (in implementation) We want to represent common values of coarse dofs as global coarse dofs, QP ::: the global coarse degrees of freedom selection matrix. Suppose there is a space Uc and operators T QP : U ! Uc Rc : Uc ! XR : U ! W ; T where Rc and R are mapping (0 − 1) matrices, related as CR = Rc QP . In implementation, Wf is decomposed into Wf = Wf∆ ⊕ WfΠ Wf∆ = functions with zero coarse dofs ) local problems on substructures WfΠ = functions given by coarse dofs & energy minimal ) ) global coarse problem. (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 14 / 49 The coarse problem A basis function from WfΠ is energy minimal subject to given values of 1 coarse degrees of freedom on the 0.8 substructure. The function is discon- 0.6 0.4 tinuous across the interfaces between 0.2 the substructures but the values of 0 coarse degrees of freedom on the dif- ferent substructures coincide. The coarse problem has the same structure as the original FE problem =) solve it approximately by one iteration of BDDC =) three-level BDDC. Apply recursively =) Multilevel BDDC. (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 15 / 49 Substructuring for the three-level method Ω 2, k Ω 1, i H2 H1 h (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 16 / 49 Multilevel BDDC Coarse problem solved by the BDDC preconditioner recursively. U q P1 E1 UI 1 ⊂ U1 ⊂ Wf1 q WfΠ1 ⊕ Wf∆1 q P2 E2 UI 2 ⊂ U2 ⊂ Wf2 q WfΠ2 ⊕ Wf∆2 q . q PL−1 EL−1 UIL−1 ⊂ UL−1 ⊂ WfL−1 q WfΠL−1 ⊕ Wf∆L−1 The leaves of the tree are the local problems and the coarse actually solved. (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 17 / 49 Condition number bound Theorem (Mandel, Soused´ık,Dohrmann (2008)) The condition number bound κ ≤ ! of Multilevel BDDC is given by 2 L−1 k(I − E`) w`ka ! = Π !` ;!` = sup `=1 kw k2 w`2(I −P`)Wf` ` a Generalizes 3-level bounds by Tu (2006, 2007) to many levels. (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 18 / 49 Adaptive method idea (Mandel, Sousedik, 2006) Condition number bound is Rayeigh quotient 2 T T k(I − E`) w`ka w` (I − E`) S` (I − E`) w` !` ≤ sup 2 = sup T kw k w S`w` w2(I −P`)Wf` ` a w`2Wf` ` Eigenvalues λ1 > λ2 > : : :, stationary points are eigenvectors. Optimal decrease of the Rayleigh quotient: make the space Wf` = ker C(I − E`) smaller by orthogonality to the dominant eigenvector u1: T add u1 to C(I − E`) This reduces the condition bound to the second eigenvalue λ2. Adding more eigenvectors eats more eigenvalues. (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 19 / 49 The condition number bound as an eigenvalue problem On the level `, define λ` by T Π` (I − E`) S` (I − E`)Π`w` = λ`Π`S`Π`w`; where S` ::: the Schur complement operator, Π` ::: the orthogonal projection in (I − P`) W` into (I − P`) Wf`. Then L−1 κ ≤ ! = Π`=1 max λ`: (Zhangjiajie, China) Adaptive Multilevel BDDC DD19, August 2009 20 / 49 Condition number bound: Heuristic indicator Solving the the global eigenvalue problems is expensive ... replace it by: local versions formulated for all pairs of adjacent substructures.
Load more

Recommended publications
  • Newton-Krylov-BDDC Solvers for Nonlinear Cardiac Mechanics
    Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics Item Type Article Authors Pavarino, L.F.; Scacchi, S.; Zampini, Stefano Citation Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics 2015 Computer Methods in Applied Mechanics and Engineering Eprint version Post-print DOI 10.1016/j.cma.2015.07.009 Publisher Elsevier BV Journal Computer Methods in Applied Mechanics and Engineering Rights NOTICE: this is the author’s version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer Methods in Applied Mechanics and Engineering, 18 July 2015. DOI:10.1016/ j.cma.2015.07.009 Download date 07/10/2021 05:34:35 Link to Item http://hdl.handle.net/10754/561071 Accepted Manuscript Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics L.F. Pavarino, S. Scacchi, S. Zampini PII: S0045-7825(15)00221-2 DOI: http://dx.doi.org/10.1016/j.cma.2015.07.009 Reference: CMA 10661 To appear in: Comput. Methods Appl. Mech. Engrg. Received date: 13 December 2014 Revised date: 3 June 2015 Accepted date: 8 July 2015 Please cite this article as: L.F. Pavarino, S. Scacchi, S. Zampini, Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics, Comput. Methods Appl. Mech. Engrg. (2015), http://dx.doi.org/10.1016/j.cma.2015.07.009 This is a PDF file of an unedited manuscript that has been accepted for publication.
    [Show full text]
  • Phd Thesis, Universit´Elibre De Bruxelles, Chauss´Eede Waterloo, 72, 1640 Rhode- St-Gen`Ese,Belgium, 2004
    Universit´eLibre de Bruxelles von Karman Institute for Fluid Dynamics Aeronautics and Aerospace Department PhD. Thesis Algorithmic Developments for a Multiphysics Framework Thomas Wuilbaut Promoter: Prof. Herman Deconinck Contact information: Thomas Wuilbaut von Karman Institute for Fluid Dynamics 72 Chauss´eede Waterloo 1640 Rhode-St-Gen`ese BELGIUM email: [email protected] webpage: http://www.vki.ac.be/~wuilbaut To Olivia and her wonderful mother... Summary v Summary In this doctoral work, we adress various problems arising when dealing with multi-physical simulations using a segregated (non-monolithic) approach. We concentrate on a few specific problems and focus on the solution of aeroelastic flutter for linear elastic structures in compressible flows, conju- gate heat transfer for re-entry vehicles including thermo-chemical reactions and finally, industrial electro-chemical plating processes which often include stiff source terms. These problems are often solved using specifically devel- oped solvers, but these cannot easily be reused for different purposes. We have therefore considered the development of a flexible and reusable software platform for the simulation of multi-physics problems. We have based this development on the COOLFluiD framework developed at the von Karman Institute in collaboration with a group of partner institutions. For the solution of fluid flow problems involving compressible flows, we have used the Finite Volume method and we have focused on the applica- tion of the method to moving and deforming computational domains using the Arbitrary Lagrangian Eulerian formulation. Validation on a series of testcases (including turbulence) is shown. In parallel, novel time integration methods have been derived from two popular time discretization methods.
    [Show full text]
  • Multigrid and Saddle-Point Preconditioners for Unfitted Finite
    Multigrid and saddle-point preconditioners for unfitted finite element modelling of inclusions Hardik Kothari∗and Rolf Krause† Institute of Computational Science, Universit`adella Svizzera italiana, Lugano, Switzerland February 17, 2021 Abstract In this work, we consider the modeling of inclusions in the material using an unfitted finite element method. In the unfitted methods, structured background meshes are used and only the underlying finite element space is modified to incorporate the discontinuities, such as inclusions. Hence, the unfitted methods provide a more flexible framework for modeling the materials with multiple inclusions. We employ the method of Lagrange multipliers for enforcing the interface conditions between the inclusions and matrix, this gives rise to the linear system of equations of saddle point type. We utilize the Uzawa method for solving the saddle point system and propose preconditioning strategies for primal and dual systems. For the dual systems, we review and compare the preconditioning strategies that are developed for FETI and SIMPLE methods. While for the primal system, we employ a tailored multigrid method specifically developed for the unfitted meshes. Lastly, the comparison between the proposed preconditioners is made through several numerical experiments. Keywords: Unfitted finite element method, multigrid method, saddle-point problem 1 Introduction In the modeling of many real-world engineering problems, we encounter material discontinuities, such as inclusions. The inclusions are found naturally in the materials, or can be artificially introduced to produce desired mechanical behavior. The finite element (FE) modeling of such inclusions requires to generate meshes that can resolve the interface between the matrix and inclusions, which can be a computationally cumbersome and expensive task.
    [Show full text]
  • Méthode De Décomposition De Domaine Avec Adaptation De
    UNIVERSIT E´ PARIS 13 No attribu´epar la biblioth`eque TH ESE` pour obtenir le grade de DOCTEUR DE L’UNIVERSIT E´ PARIS 13 Discipline: Math´ematiques Appliqu´ees Laboratoire d’accueil: ONERA - Le centre fran¸cais de recherche a´erospatiale Pr´esent´ee et soutenue publiquement le 19 d´ecembre 2014 par Oana Alexandra CIOBANU Titre M´ethode de d´ecomposition de domaine avec adaptation de maillage en espace-temps pour les ´equations d’Euler et de Navier–Stokes devant le jury compos´ede: Fran¸cois Dubois Rapporteur Laurence Halpern Directrice de th`ese Rapha`ele Herbin Rapporteure Xavier Juvigny Examinateur Olivier Lafitte Examinateur Juliette Ryan Encadrante UNIVERSITY PARIS 13 THESIS Presented for the degree of DOCTEUR DE L’UNIVERSIT E´ PARIS 13 In Applied Mathematics Hosting laboratory: ONERA - The French Aerospace Lab presented for public discussion on 19 decembre 2014 by Oana Alexandra CIOBANU Subject Adaptive Space-Time Domain Decomposition Methods for Euler and Navier–Stokes Equations Jury: Fran¸cois Dubois Reviewer Laurence Halpern Supervisor Rapha`ele Herbin Reviewer Xavier Juvigny Examiner Olivier Lafitte Examiner Juliette Ryan Supervisor newpage Remerciements Tout d’abord, je remercie grandement Juliette Ryan pour avoir accept´e d’ˆetre mon encad- rante de stage puis mon encadrante de th`ese. Pendant plus de trois ans, elle m’a fait d´ecouvrir mon m´etier de jeune chercheuse avec beaucoup de patience et de professionnalisme. Elle m’a soutenue, elle a ´et´ed’une disponibilit´eet d’une ´ecoute extraordinaires, tout en sachant ˆetre rigoureuse et exigeante avec moi comme avec elle-mˆeme. Humainement, j’ai beaucoup appr´eci´e la relation d’amiti´eque nous avons entretenue, le climat de confiance que nous avons maintenu et les discussions extra-math´ematiques que nous avons pu avoir et qui ont renforc´ele lien que nous avions.
    [Show full text]
  • A FETI-DP Method for the Parallel Iterative Solution of Indefinite And
    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng (in press) Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1282 A FETI-DP method for the parallel iterative solution of indefinite and complex-valued solid and shell vibration problems Charbel Farhat1,∗,†, Jing Li2 and Philip Avery1 1Department of Mechanical Engineering and Institute for Computational and Mathematical Engineering, Stanford University, Building 500, Stanford, CA 94305-3035, U.S.A. 2Department of Mathematical Sciences, Kent State University, Kent, OH 44242, U.S.A. SUMMARY The dual-primal finite element tearing and interconnecting (FETI-DP) domain decomposition method (DDM) is extended to address the iterative solution of a class of indefinite problems of the form K − 2M u = f, and a class of complex problems of the form K − 2M + iD u = f, where K, M, and D are three real symmetric matrices arising from the finite element discretization of solid and shell dynamic problems, i is the imaginary complex number, and is a real positive number. A key component of this extension is a new coarse problem based on the free-space solutions of Navier’s equations of motion. These solutions are waves, and therefore the resulting DDM is reminiscent of the FETI-H method. For this reason, it is named here the FETI-DPH method. For a practically large range, FETI-DPH is shown numerically to be scalable with respect to all of the problem size, substructure size, and number of substructures. The CPU performance of this iterative solver is illustrated on a 40-processor computing system with the parallel solution, for various ranges, of several large-scale, indefinite, or complex-valued systems of equations associated with shifted eigenvalue and forced frequency response structural dynamics problems.
    [Show full text]
  • An Extension of the FETI Domain Decomposition Method for Incompressible and Nearly Incompressible Problems Benoit Vereecke, Henri Bavestrello, David Dureisseix
    An extension of the FETI domain decomposition method for incompressible and nearly incompressible problems Benoit Vereecke, Henri Bavestrello, David Dureisseix To cite this version: Benoit Vereecke, Henri Bavestrello, David Dureisseix. An extension of the FETI domain decomposi- tion method for incompressible and nearly incompressible problems. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2003, 192 (31-32), pp.3409-3429. 10.1016/S0045-7825(03)00313- X. hal-00141163 HAL Id: hal-00141163 https://hal.archives-ouvertes.fr/hal-00141163 Submitted on 12 Apr 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. An extension of the FETI domain decomposition method for incompressible and nearly incompressible problems B. Vereecke,∗ H. Bavestrello,† D. Dureisseix‡ Abstract Incompressible and nearly incompressible problems are treated herein with a mixed finite element formulation in order to avoid ill-conditioning that prevents accuracy in pressure estimation and lack of convergence for iterative solution algorithms. A multilevel dual domain decomposition method is then chosen as an iterative algorithm: the original FETI and FETI-DP methods are extended to deal with such problems, when the dis- cretization of the pressure field is discontinuous throughout the elements.
    [Show full text]
  • Preconditioning the Coarse Problem of BDDC Methods—Three-Level, Algebraic Multigrid, and Vertex-Based Preconditioners
    Electronic Transactions on Numerical Analysis. Volume 51, pp. 432–450, 2019. ETNA Kent State University and Copyright c 2019, Kent State University. Johann Radon Institute (RICAM) ISSN 1068–9613. DOI: 10.1553/etna_vol51s432 PRECONDITIONING THE COARSE PROBLEM OF BDDC METHODS— THREE-LEVEL, ALGEBRAIC MULTIGRID, AND VERTEX-BASED PRECONDITIONERS∗ AXEL KLAWONNyz, MARTIN LANSERyz, OLIVER RHEINBACHx, AND JANINE WEBERy Abstract. A comparison of three Balancing Domain Decomposition by Constraints (BDDC) methods with an approximate coarse space solver using the same software building blocks is attempted for the first time. The comparison is made for a BDDC method with an algebraic multigrid preconditioner for the coarse problem, a three-level BDDC method, and a BDDC method with a vertex-based coarse preconditioner. It is new that all methods are presented and discussed in a common framework. Condition number bounds are provided for all approaches. All methods are implemented in a common highly parallel scalable BDDC software package based on PETSc to allow for a simple and meaningful comparison. Numerical results showing the parallel scalability are presented for the equations of linear elasticity. For the first time, this includes parallel scalability tests for a vertex-based approximate BDDC method. Key words. approximate BDDC, three-level BDDC, multilevel BDDC, vertex-based BDDC AMS subject classifications. 68W10, 65N22, 65N55, 65F08, 65F10, 65Y05 1. Introduction. During the last decade, approximate variants of the BDDC (Balanc- ing Domain Decomposition by Constraints) and FETI-DP (Finite Element Tearing and Interconnecting-Dual-Primal) methods have become popular for the solution of various linear and nonlinear partial differential equations [1,8,9, 12, 14, 15, 17, 19, 21, 24, 25].
    [Show full text]
  • FETI-DP Domain Decomposition Method
    ECOLE´ POLYTECHNIQUE FED´ ERALE´ DE LAUSANNE Master Project FETI-DP Domain Decomposition Method Supervisors: Author: Davide Forti Christoph Jaggli¨ Dr. Simone Deparis Prof. Alfio Quarteroni A thesis submitted in fulfilment of the requirements for the degree of Master in Mathematical Engineering in the Chair of Modelling and Scientific Computing Mathematics Section June 2014 Declaration of Authorship I, Christoph Jaggli¨ , declare that this thesis titled, 'FETI-DP Domain Decomposition Method' and the work presented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree at this University. Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly at- tributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. I have acknowledged all main sources of help. Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself. Signed: Date: ii ECOLE´ POLYTECHNIQUE FED´ ERALE´ DE LAUSANNE Abstract School of Basic Science Mathematics Section Master in Mathematical Engineering FETI-DP Domain Decomposition Method by Christoph Jaggli¨ FETI-DP is a dual iterative, nonoverlapping domain decomposition method. By a Schur complement procedure, the solution of a boundary value problem is reduced to solving a symmetric and positive definite dual problem in which the variables are directly related to the continuity of the solution across the interface between the subdomains.
    [Show full text]
  • Feti-Dp Method for Dg Discretization of Elliptic Problems with Discontinuous Coefficients
    FETI-DP METHOD FOR DG DISCRETIZATION OF ELLIPTIC PROBLEMS WITH DISCONTINUOUS COEFFICIENTS MAKSYMILIAN DRYJA∗ AND MARCUS SARKIS∗† Abstract. In this paper a discontinuous Galerkin (DG) discretization of an elliptic two- dimensional problem with discontinuous coefficients is considered. The problem is posed on a polyg- onal region Ω which is a union of disjoint polygonals Ωi of diameter O(Hi) and forms a geometrically conforming partition of Ω. The discontinuities of the coefficients are assumed to occur only across ∂Ωi. Inside of each substructure Ωi, a conforming finite element space on a quasiuniform triangula- tion with triangular elements and mesh size O(hi) is introduced. To handle the nonmatching meshes across ∂Ωi, a discontinuous Galerkin discretization is considered. For solving the resulting discrete problem, a FETI-DP method is designed and analyzed. It is established that the condition number 2 of the method is estimated by C(1 + maxi log Hi/hi) with a constant C independent of hi, Hi and the jumps of the coefficients. The method is well suited for parallel computations and it can be straightforwardly extended to three-dimensional problems. Key words. Interior penalty discretization, discontinuous Galerkin, elliptic problems with discontinuous coefficients. finite element method, FETI-DP algorithms, preconditioners AMS subject classifications. 65F10, 65N20, 65N30 1. Introduction. In this paper a discontinuous Galerkin (DG) approximation of an elliptic problem with discontinuous coefficients is considered. The problem is posed on a polygonal region Ω which is a union of disjoint polygonal subregions Ωi of diameter O(Hi) and forms a geometrically partition of Ω, i.e., for i 6= j, ∂Ωi ∩ ∂Ωj is empty or is a common corner or edge of ∂Ωi and ∂Ωj, where an edge means a curve of continuous intervals.
    [Show full text]
  • Domain Decomposition Methods for Problems in H(Curl)
    Domain Decomposition Methods for Problems in H (curl) by Juan Gabriel Calvo A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics New York University September 2015 Professor Olof B. Widlund ©Juan Gabriel Calvo All rights reserved, 2015 Dedication To my family. iv Acknowledgements First, my deepest gratitude goes to my advisor Olof Widlund. I thank him profoundly for his direction, guidance, support and advice during four years. I would also like to thank the rest of my committee: Professors Berger, Good- man, O'Neil and Stadler. In addition, I also thank Dr. Clark Dohrmann of the SANDIA-Albuquerque laboratories for his help and comments throughout my re- search. Finally I thank my Alma Mater, Universidad de Costa Rica, for the support during my academic education at NYU. v Abstract Two domain decomposition methods for solving vector field problems posed in H(curl) and discretized with N´ed´elecfinite elements are considered. These finite elements are conforming in H(curl). A two-level overlapping Schwarz algorithm in two dimensions is analyzed, where the subdomains are only assumed to be uniform in the sense of Peter Jones. The coarse space is based on energy minimization and its dimension equals the number of interior subdomain edges. Local direct solvers are based on the overlapping subdomains. The bound for the condition number depends only on a few geometric parameters of the decomposition. This bound is independent of jumps in the coefficients across the interface between the subdomains for most of the different cases considered.
    [Show full text]
  • Multigrid Solvers for Immersed Finite Element Methods and Immersed Isogeometric Analysis
    Computational Mechanics https://doi.org/10.1007/s00466-019-01796-y ORIGINAL PAPER Multigrid solvers for immersed finite element methods and immersed isogeometric analysis F. de Prenter1,3 · C. V. Verhoosel1 · E. H. van Brummelen1 · J. A. Evans2 · C. Messe2,4 · J. Benzaken2,5 · K. Maute2 Received: 26 March 2019 / Accepted: 10 November 2019 © The Author(s) 2019 Abstract Ill-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeo- metric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system matrix, which generally degrades efficiency and robustness of iterative solvers. In this contribution we investigate the spectral prop- erties of immersed finite element systems treated by Schwarz-type methods, to establish the suitability of these as smoothers in a multigrid method. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides mesh-independent and cut-element-independent convergence rates. This preconditioning technique is applicable to higher-order discretizations, and enables solving large-scale immersed systems at a computational cost that scales linearly with the number of degrees of freedom. The performance of the preconditioner is demonstrated for conventional Lagrange basis functions and for isogeometric discretizations with both uniform B-splines and locally refined approximations based on truncated hierarchical B-splines. Keywords Immersed finite element method · Fictitious domain method · Iterative solver · Preconditioner · Multigrid 1 Introduction [14–21], scan based analysis [22–27] and topology optimiza- tion, e.g., [28–34]. Immersed methods are useful tools to avoid laborious and An essential aspect of finite element methods and iso- computationally expensive procedures for the generation of geometric analysis is the computation of the solution to a body-fitted finite element discretizations or analysis-suitable system of equations.
    [Show full text]
  • Using Algebraic Multigrid in Inexact BDDC Domain Decomposition Methods
    Using Algebraic Multigrid in Inexact BDDC Domain Decomposition Methods Axel Klawonn, Martin Lanser, and Oliver Rheinbach 1 Introduction Traditionally, domain decomposition methods use sparse direct solvers as building blocks, i.e., to solve local subdomain problems and/or the coarse problem. Often, the sparse direct solvers can be replaced by spectrally equivalent preconditioners without loss of convergence speed. In FETI-DP and BDDC domain decomposition methods, such approaches have first been introduced in [9, 8, 4], and have since then successfully been used in large parallel codes [6, 1]. 2 An Inexact BDDC Method 2.1 A BDDC Preconditioner for the Assembled System Let us briefly describe the BDDC preconditioner which can directly be applied to a linear system Au = b (1) arising from a finite element discretization of a partial differential equation on a d computational domain W ⊂ R ; d = 2;3. The variant discussed here was first intro- duced in [9]. Let Wi; i = 1;:::;N; be a nonoverlapping domain decomposition of SN W such that W = i=1 W i. Each subdomain Wi is discretized using finite elements, Axel Klawonn, Martin Lanser Mathematisches Institut, Universitat¨ zu Koln,¨ Weyertal 86-90, 50931 Koln,¨ Germany, e-mail: faxel.klawonn,[email protected] Oliver Rheinbach Institut fur¨ Numerische Mathematik und Optimierung, Fakultat¨ fur¨ Mathematik und Infor- matik, Technische Universitat¨ Bergakademie Freiberg, Akademiestr. 6, 09596 Freiberg, e-mail: [email protected] 1 2 Axel Klawonn, Martin Lanser, and Oliver Rheinbach the corresponding local finite element spaces are denoted by Wi; i = 1;:::;N, and the product space is defined by W = W1 ×:::×WN.
    [Show full text]