A Hybrid FETI-DP Method for Non-Smooth Random Partial Differential Equations
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38. a Preconditioner for the Schur Complement Domain Decomposition Method J.-M
Fourteenth International Conference on Domain Decomposition Methods Editors: Ismael Herrera , David E. Keyes, Olof B. Widlund, Robert Yates c 2003 DDM.org 38. A preconditioner for the Schur complement domain decomposition method J.-M. Cros 1 1. Introduction. This paper presents a preconditioner for the Schur complement domain decomposition method inspired by the dual-primal FETI method [4]. Indeed the proposed method enforces the continuity of the preconditioned gradient at cross-points di- rectly by a reformulation of the classical Neumann-Neumann preconditioner. In the case of elasticity problems discretized by finite elements, the degrees of freedom corresponding to the cross-points coming from domain decomposition, in the stiffness matrix, are separated from the rest. Elimination of the remaining degrees of freedom results in a Schur complement ma- trix for the cross-points. This assembled matrix represents the coarse problem. The method is not mathematically optimal as shown by numerical results but its use is rather economical. The paper is organized as follows: in sections 2 and 3, the Schur complement method and the formulation of the Neumann-Neumann preconditioner are briefly recalled to introduce the notations. Section 4 is devoted to the reformulation of the Neumann-Neumann precon- ditioner. In section 5, the proposed method is compared with other domain decomposition methods such as generalized Neumann-Neumann algorithm [7][9], one-level FETI method [5] and dual-primal FETI method. Performances on a parallel machine are also given for structural analysis problems. 2. The Schur complement domain decomposition method. Let Ω denote the computational domain of an elasticity problem. -
A High-Order Boris Integrator
A high-order Boris integrator Mathias Winkela,∗, Robert Speckb,a, Daniel Ruprechta aInstitute of Computational Science, University of Lugano, Switzerland. bJ¨ulichSupercomputing Centre, Forschungszentrum J¨ulich,Germany. Abstract This work introduces the high-order Boris-SDC method for integrating the equations of motion for electrically charged particles in an electric and magnetic field. Boris-SDC relies on a combination of the Boris-integrator with spectral deferred corrections (SDC). SDC can be considered as preconditioned Picard iteration to com- pute the stages of a collocation method. In this interpretation, inverting the preconditioner corresponds to a sweep with a low-order method. In Boris-SDC, the Boris method, a second-order Lorentz force integrator based on velocity-Verlet, is used as a sweeper/preconditioner. The presented method provides a generic way to extend the classical Boris integrator, which is widely used in essentially all particle-based plasma physics simulations involving magnetic fields, to a high-order method. Stability, convergence order and conserva- tion properties of the method are demonstrated for different simulation setups. Boris-SDC reproduces the expected high order of convergence for a single particle and for the center-of-mass of a particle cloud in a Penning trap and shows good long-term energy stability. Keywords: Boris integrator, time integration, magnetic field, high-order, spectral deferred corrections (SDC), collocation method 1. Introduction Often when modeling phenomena in plasma physics, for example particle dynamics in fusion vessels or particle accelerators, an externally applied magnetic field is vital to confine the particles in the physical device [1, 2]. In many cases, such as instabilities [3] and high-intensity laser plasma interaction [4], the magnetic field even governs the microscopic evolution and drives the phenomena to be studied. -
Domain Decomposition Solvers (FETI) Divide Et Impera
Domain Decomposition solvers (FETI) a random walk in history and some current trends Daniel J. Rixen Technische Universität München Institute of Applied Mechanics www.amm.mw.tum.de [email protected] 8-10 October 2014 39th Woudschoten Conference, organised by the Werkgemeenschap Scientific Computing (WSC) 1 Divide et impera Center for Aerospace Structures CU, Boulder wikipedia When splitting the problem in parts and asking different cpu‘s (or threads) to take care of subproblems, will the problem be solved faster ? FETI, Primal Schur (Balancing) method around 1990 …….. basic methods, mesh decomposer technology 1990-2001 .…….. improvements •! preconditioners, coarse grids •! application to Helmholtz, dynamics, non-linear ... Here the concepts are outlined using some mechanical interpretation. For mathematical details, see lecture of Axel Klawonn. !"!! !"#$%&'()$&'(*+*',-%-.&$(*&$-%'#$%-'(*-/$0%1*()%-).1*,2%()'(%()$%,.,3$4*-($,+$% .5%'%-.6"(*.,%*&76*$-%'%6.2*+'6%+.,(8'0*+(*.,9%1)*6$%$,2*,$$#-%&*2)(%+.,-*0$#%'% ,"&$#*+'6%#$-"6(%'-%()$%.,6:%#$'-.,';6$%2.'6<%% ="+)%.,$%-*0$0%>*$1-%-$$&%(.%#$?$+(%)"&',%6*&*('(*.,-%#'()$#%()',%.;@$+(*>$%>'6"$-<%% A,%*(-$65%&'()$&'(*+-%*-%',%*,0*>*-*;6$%.#2',*-&%",*(*,2%()$.#$(*+'6%+.,($&76'(*.,% ',0%'+(*>$%'776*+'(*.,<%%%%% %%%%%%% R. Courant %B % in Variational! Methods for the solution of problems of equilibrium and vibrations Bulletin of American Mathematical Society, 49, pp.1-23, 1943 Here the concepts are outlined using some mechanical interpretation. For mathematical details, see lecture of Axel Klawonn. Content -
Strong Form Meshfree Collocation Method for Higher Order and Nonlinear Pdes in Engineering Applications
University of Colorado, Boulder CU Scholar Civil Engineering Graduate Theses & Dissertations Civil, Environmental, and Architectural Engineering Spring 1-1-2019 Strong Form Meshfree Collocation Method for Higher Order and Nonlinear Pdes in Engineering Applications Andrew Christian Beel University of Colorado at Boulder, [email protected] Follow this and additional works at: https://scholar.colorado.edu/cven_gradetds Part of the Civil Engineering Commons, and the Partial Differential Equations Commons Recommended Citation Beel, Andrew Christian, "Strong Form Meshfree Collocation Method for Higher Order and Nonlinear Pdes in Engineering Applications" (2019). Civil Engineering Graduate Theses & Dissertations. 471. https://scholar.colorado.edu/cven_gradetds/471 This Thesis is brought to you for free and open access by Civil, Environmental, and Architectural Engineering at CU Scholar. It has been accepted for inclusion in Civil Engineering Graduate Theses & Dissertations by an authorized administrator of CU Scholar. For more information, please contact [email protected]. Strong Form Meshfree Collocation Method for Higher Order and Nonlinear PDEs in Engineering Applications Andrew Christian Beel B.S., University of Colorado Boulder, 2019 A thesis submitted to the Faculty of the Graduate School of the University of Colorado Boulder in partial fulfillment of the requirement for the degree of Master of Science Department of Civil, Environmental and Architectural Engineering 2019 This thesis entitled: Strong Form Meshfree Collocation Method for Higher Order and Nonlinear PDEs in Engineering Applications written by Andrew Christian Beel has been approved for the Department of Civil, Environmental and Architectural Engineering Dr. Jeong-Hoon Song (Committee Chair) Dr. Ronald Pak Dr. Victor Saouma Dr. Richard Regueiro Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. -
Numerical Solution of Saddle Point Problems
Acta Numerica (2005), pp. 1–137 c Cambridge University Press, 2005 DOI: 10.1017/S0962492904000212 Printed in the United Kingdom Numerical solution of saddle point problems Michele Benzi∗ Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322, USA E-mail: [email protected] Gene H. Golub† Scientific Computing and Computational Mathematics Program, Stanford University, Stanford, California 94305-9025, USA E-mail: [email protected] J¨org Liesen‡ Institut f¨ur Mathematik, Technische Universit¨at Berlin, D-10623 Berlin, Germany E-mail: [email protected] We dedicate this paper to Gil Strang on the occasion of his 70th birthday Large linear systems of saddle point type arise in a wide variety of applica- tions throughout computational science and engineering. Due to their indef- initeness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems. ∗ Supported in part by the National Science Foundation grant DMS-0207599. † Supported in part by the Department of Energy of the United States Government. ‡ Supported in part by the Emmy Noether Programm of the Deutsche Forschungs- gemeinschaft. 2 M. Benzi, G. H. Golub and J. Liesen CONTENTS 1 Introduction 2 2 Applications leading to saddle point problems 5 3 Properties of saddle point matrices 14 4 Overview of solution algorithms 29 5 Schur complement reduction 30 6 Null space methods 32 7 Coupled direct solvers 40 8 Stationary iterations 43 9 Krylov subspace methods 49 10 Preconditioners 59 11 Multilevel methods 96 12 Available software 105 13 Concluding remarks 107 References 109 1. -
Workshop High-Order Numerical Approximation for Partial Differential Equations
Workshop High-Order Numerical Approximation for Partial Differential Equations Hausdorff Center for Mathematics University of Bonn February 6-10, 2012 Organizers Alexey Chernov (University of Bonn) Christoph Schwab (ETH Zurich) Program overview Time Monday Tuesday Wednesday Thursday Friday 8:00 Registration 8:40 Opening 9:00 Stephan Sloan Ainsworth Demkowicz Buffa 10:00 Canuto Nobile Sherwin Houston Beir~ao da Veiga 10:30 Reali 11:00 Coffee break 11:30 Quarteroni Gittelson Sch¨oberl Sch¨otzau Sangalli 12:00 Bespalov 12:30 Lunch break 14:30 Dauge Xiu Wihler D¨uster Excursions: 15:00 Yosibash Arithmeum 15:30 Melenk Litvinenko Hesthaven or Coffee break 16:00 Maischak Rank Botanic 16:30 Coffee break Coffee break Gardens 17:00 Maday Hackbusch Hiptmair {18:00 Free Time 20:00 Dinner 2 Detailed program Monday, 9:00{9:55 Ernst P. Stephan hp-adaptive DG-FEM for Parabolic Obstacle Problems Monday, 10:00{10:55 Claudio Canuto Adaptive Fourier-Galerkin Methods Monday, 11:30{12:25 Alfio Quarteroni Discontinuous approximation of elastodynamics equations Monday, 14:30{15:25 Monique Dauge Weighted analytic regularity in polyhedra Monday, 15:30{16:25 Jens Markus Melenk Stability and convergence of Galerkin discretizations of the Helmholtz equation Monday, 17:00{17:55 Yvon Maday A generalized empirical interpolation method based on moments and application Tuesday, 9:00{9:55 Ian H. Sloan PDE with random coefficients as a problem in high dimensional integration Tuesday, 10:00{10:55 Fabio Nobile Stochastic Polynomial approximation of PDEs with random coefficients Tuesday, -
Facts from Linear Algebra
Appendix A Facts from Linear Algebra Abstract We introduce the notation of vector and matrices (cf. Section A.1), and recall the solvability of linear systems (cf. Section A.2). Section A.3 introduces the spectrum σ(A), matrix polynomials P (A) and their spectra, the spectral radius ρ(A), and its properties. Block structures are introduced in Section A.4. Subjects of Section A.5 are orthogonal and orthonormal vectors, orthogonalisation, the QR method, and orthogonal projections. Section A.6 is devoted to the Schur normal form (§A.6.1) and the Jordan normal form (§A.6.2). Diagonalisability is discussed in §A.6.3. Finally, in §A.6.4, the singular value decomposition is explained. A.1 Notation for Vectors and Matrices We recall that the field K denotes either R or C. Given a finite index set I, the linear I space of all vectors x =(xi)i∈I with xi ∈ K is denoted by K . The corresponding square matrices form the space KI×I . KI×J with another index set J describes rectangular matrices mapping KJ into KI . The linear subspace of a vector space V spanned by the vectors {xα ∈V : α ∈ I} is denoted and defined by α α span{x : α ∈ I} := aαx : aα ∈ K . α∈I I×I T Let A =(aαβ)α,β∈I ∈ K . Then A =(aβα)α,β∈I denotes the transposed H matrix, while A =(aβα)α,β∈I is the adjoint (or Hermitian transposed) matrix. T H Note that A = A holds if K = R . Since (x1,x2,...) indicates a row vector, T (x1,x2,...) is used for a column vector. -
ICES REPORT 14-04 Collocation on Hp Finite Element Meshes
ICES REPORT 14-04 February 2014 Collocation On Hp Finite Element Meshes: Reduced Quadrature Perspective, Cost Comparison With Standard Finite Elements, And Explicit Structural Dynamics by Dominik Schillinger, John a. Evans, Felix Frischmann, Rene R. Hiemstra, Ming-Chen Hsu, Thomas J.R. Hughes The Institute for Computational Engineering and Sciences The University of Texas at Austin Austin, Texas 78712 Reference: Dominik Schillinger, John a. Evans, Felix Frischmann, Rene R. Hiemstra, Ming-Chen Hsu, Thomas J.R. Hughes, "Collocation On Hp Finite Element Meshes: Reduced Quadrature Perspective, Cost Comparison With Standard Finite Elements, And Explicit Structural Dynamics," ICES REPORT 14-04, The Institute for Computational Engineering and Sciences, The University of Texas at Austin, February 2014. Collocation on hp finite element meshes: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics Dominik Schillingera,b,∗, John A. Evansc, Felix Frischmannd, Ren´eR. Hiemstrab, Ming-Chen Hsue, Thomas J.R. Hughesb aDepartment of Civil Engineering, University of Minnesota, Twin Cities, USA bInstitute for Computational Engineering and Sciences, The University of Texas at Austin, USA cAerospace Engineering Sciences, University of Colorado Boulder, USA dLehrstuhl f¨ur Computation in Engineering, Technische Universit¨at M¨unchen, Germany eDepartment of Mechanical Engineering, Iowa State University, Ames, USA Abstract We demonstrate the potential of collocation methods for efficient higher-order analysis on standard nodal finite element meshes. We focus on a collocation method that is variation- ally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. -
The Numerical Solution of Helmholtz Equation Using Modified Wavelet Multigrid Method
International Journal of Modern Mathematical Sciences, 2020, 18(1): 76-91 International Journal of Modern Mathematical Sciences ISSN:2166-286X Journal homepage:www.ModernScientificPress.com/Journals/ijmms.aspx Florida, USA Article The Numerical Solution of Helmholtz Equation Using Modified Wavelet Multigrid Method S. C. Shiralashetti1, A. B. Deshi2,*, M. H. Kantli3 1Department of Mathematics, Karnatak University, Dharwad, India-580003 2Department of Mathematics, KLECET, Chikodi, India-591201 3Department of Mathematics, BVV’s BGMIT, Mudhol, India-587313 *Author to whom correspondence should be addressed; E-Mail: [email protected] Article history: Received 11 June 2020, Revised 23 July 2020, Accepted 1 September 2020, Published 14 September 2020. Abstract: This paper presents a modified wavelet multigrid technique for solving elliptic type partial differential equations namely Helmholtz equation. The solution is first obtained on the coarser grid points, and then it is refined by obtaining higher accuracy by increasing the level of resolution. The implementation of the classical numerical methods has been found to involve some difficulty to observe fast convergence in low computational time. To overcome this, we have proposed modified wavelet multigrid method using wavelet intergrid operators similar to classical intergrid operators. Some of the numerical test problems are presented to demonstrate the applicability and attractiveness of the present implementation. Keywords: Wavelet multigrid; Helmholtz equation; Intergrid operators; Numerical methods. Mathematics Subject Classification: 65T60, 97N40, 35J25 1. Introduction The mathematical modelling of engineering problems usually leads to sets of partial differential equations and their boundary conditions. There are several applications of elliptic partial differential equations (EPDEs) in science and engineering. Many physical processes can be modeled using EPDEs. -
A Local Hybrid Surrogate‐Based Finite Element Tearing Interconnecting
Received: 6 January 2020 Revised: 5 October 2020 Accepted: 17 October 2020 DOI: 10.1002/nme.6571 RESEARCH ARTICLE A local hybrid surrogate-based finite element tearing interconnecting dual-primal method for nonsmooth random partial differential equations Martin Eigel Robert Gruhlke Research Group “Nonlinear Optimization and Inverse Problems”, Weierstrass Abstract Institute, Berlin, Germany A domain decomposition approach for high-dimensional random partial differ- ential equations exploiting the localization of random parameters is presented. Correspondence Robert Gruhlke, Weierstrass Institute, To obtain high efficiency, surrogate models in multielement representations Mohrenstrasse 39 D-10117, Berlin, in the parameter space are constructed locally when possible. The method Germany. makes use of a stochastic Galerkin finite element tearing interconnecting Email: [email protected] dual-primal formulation of the underlying problem with localized representa- Funding information tions of involved input random fields. Each local parameter space associated to Deutsche Forschungsgemeinschaft, Grant/Award Number: DFGHO1947/10-1 a subdomain is explored by a subdivision into regions where either the para- metric surrogate accuracy can be trusted or where instead one has to resort to Monte Carlo. A heuristic adaptive algorithm carries out a problem-dependent hp-refinement in a stochastic multielement sense, anisotropically enlarging the trusted surrogate region as far as possible. This results in an efficient global parameter to solution sampling scheme making use of local parametric smooth- ness exploration for the surrogate construction. Adequately structured problems for this scheme occur naturally when uncertainties are defined on subdomains, for example, in a multiphysics setting, or when the Karhunen–Loève expan- sion of a random field can be localized. -
Partial Differential Equations
Next: Using Matlab Up: Numerical Analysis for Chemical Previous: Ordinary Differential Equations Subsections z Finite Difference: Elliptic Equations { The Laplace Equations { Solution Techniques { Boundary Conditions { The Control Volume Approach z Finite Difference: Parabolic Equations { The Heat Conduction Equation { Explicit Methods { A Simple Implicit Method { The Crank-Nicholson Method z Finite Element Method { Calculus of variation { Example: The shortest distance between two points { The Rayleigh-Ritz Method { The Collocation and Galerkin Method { Finite elements for ordinary-differential equations z Engineering Applications: Partial Differential Equations Partial Differential Equations An equation involving partial derivatives of an unknown function of two or more independent variables is called a partial differential equation, PDE. The order of a PDE is that of the highest-order partial derivative appearing in the equation. A general linear second-order differential equation is (8.1) Depending on the values of the coefficients of the second-derivative terms eq. (8.1) can be classified int one of three categories. z : Elliptic Laplace equation(steady state with two spatial dimensions) z : Parabolic Heat conduction equation(time variable with one spatial dimension) z : Hyperbolic Wave equation(time variable with one spatial dimension) Finite Difference: Elliptic Equations Elliptic equations in engineering are typically used to characterize steady-state, boundary-value problems. The Laplace Equations z The Laplace equation z The Poisson equation Solution Techniques z The Laplacian Difference Equation : use central difference based on the grid scheme and Substituting these equations into the Laplace equation gives For the square grid, , and by collecting terms This relationship, which holds for all interior point on the plate, is referred to as the Laplacian difference equation. -
65 Numerical Analysis
e Q (e t o 5 M SectionsSet 1Q (Section 65)MR September 2012 65 NUMERICAL ANALYSIS MR2918625 65-06 FRecent advances in scientific computing and matrix analysis. Proceedings of the International Workshop held at the University of Macau, Macau, December 28{30, 2009. Edited by Xiao-Qing Jin, Hai-Wei Sun and Seak-Weng Vong. International Press, Somerville, MA; Higher Education Press, Beijing, 2011. xii+126 pp. ISBN 978-1-57146-202-2 Contents: Zheng-jian Bai and Xiao-qing Jin [Xiao Qing Jin1], A note on the Ulm-like method for inverse eigenvalue problems (1{7) MR2908437; Che-man Cheng [Che-Man Cheng], Kin-sio Fong [Kin-Sio Fong] and Io-kei Lok [Io-Kei Lok], Another proof for commutators with maximal Frobenius norm (9{14) MR2908438; Wai-ki Ching [Wai-Ki Ching] and Dong-mei Zhu [Dong Mei Zhu1], On high-dimensional Markov chain models for categorical data sequences with applications (15{34) MR2908439; Yan-nei Law [Yan Nei Law], Hwee-kuan Lee [Hwee Kuan Lee], Chao-qiang Liu [Chaoqiang Liu] and Andy M. Yip, An additive variational model for image segmentation (35{48) MR2908440; Hai-yong Liao [Haiyong Liao] and Michael K. Ng, Total variation image restoration with automatic selection of regularization parameters (49{59) MR2908441; Franklin T. Luk and San-zheng Qiao [San Zheng Qiao], Matrices and the LLL algorithm (61{69) MR2908442; Mila Nikolova, Michael K. Ng and Chi-pan Tam [Chi-Pan Tam], A fast nonconvex nonsmooth minimization method for image restoration and reconstruction (71{83) MR2908443; Gang Wu [Gang Wu1], Eigenvalues of certain augmented complex stochastic matrices with applications to PageRank (85{92) MR2908444; Yan Xuan and Fu-rong Lin, Clenshaw-Curtis-rational quadrature rule for Wiener-Hopf equations of the second kind (93{110) MR2908445; Man-chung Yeung [Man-Chung Yeung], On the solution of singular systems by Krylov subspace methods (111{116) MR2908446; Qi- fang Yu, San-zheng Qiao [San Zheng Qiao] and Yi-min Wei, A comparative study of the LLL algorithm (117{126) MR2908447.