DOCSLIB.ORG

Multigrid and Saddle-Point Preconditioners for Unfitted Finite

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. In order to avoid the need of generating such fitted meshes, the extended finite element method (XFEM) was introduced [13]. The XFEM method can be categorized as an unfitted finite element method. For the unfitted FE methods, it is not necessary to have the meshes that resolve the interfaces exactly. In this framework, structured uniform meshes are typically used and the associated FE spaces are enriched to capture the interface information. There are multiple ways for enriching the FE spaces and for enforcing the constraints on the interfaces. Here, we employ the method of Lagrange multipliers to enforce the interface conditions. The method of Lagrange multipliers is quite robust, but it gives rise to the mixed formulation and requires the solution of the saddle point type linear system of equations. arXiv:2102.08112v1 [math.NA] 16 Feb 2021 In this work, we model a domain with multiple inclusions using a variant of the unfitted FE method. We briefly introduce the unfitted discretization methods and then discuss the solution strategies for solving the arising saddle-point systems. In particular, we present a survey on the available preconditioning strategies for solving the primal and dual systems. 2 Model problem 2 We assume a domain Ω 2 R with the Lipschitz continuous boundary @Ω. This domain Ω is assumed to be decomposed into 1 2 n distinct non-overlapping parts, the matrix ΩM and n inclusions fΩI ; ΩI ;:::; ΩI g. The inclusions are assumed to be completely 1 2 n embedded in the matrix domain ΩM . Each inclusion is separated from the domain ΩM by interfaces fΓ ; Γ ;:::; Γ g. The interfaces associated with the inclusions are assumed to be sufficiently smooth. Thus, the domain is defined as, Ω = Sn i i i (ΩI [ Γ ) [ ΩM . The boundary @Ω is decomposed into Dirichlet and Neumann boundaries given as @ΩD and @ΩN , respectively. We assume that the domain is subjected to volume forces f and the traction/surface forces tN . Under the ∗[email protected] †[email protected] 1 14th World Congress on Computational14th World Mechanics Congress (WCCM) on Computational14th World Mechanics Congress (WCCM) on Computational14th World Mechanics Congress (WCCM) on Computational Mechanics (WCCM) ECCOMAS Congress 2020 ECCOMAS Congress 2020 ECCOMAS Congress 2020 ECCOMAS Congress 2020 Virtual Congress: 11 – 15 JanuaryVirtual 2021 Congress: 11 – 15 JanuaryVirtual 2021 Congress: 11 – 15 JanuaryVirtual 2021 Congress: 11 – 15 January 2021 F. Chinesta, R. Abgrall, O. Allix andF. Chinesta, M. Kaliske R. (Eds) Abgrall, O. Allix andF. Chinesta, M. Kaliske R. (Eds) Abgrall, O. Allix andF. Chinesta, M. Kaliske R. (Eds) Abgrall, O. Allix and M. Kaliske (Eds) 1 1 1 1 i (a) Inclusions in a matrix (b) An unfitted mesh Teh (c) Th;M encapsulating the (d) Th;I encapsulating the in- i matrix ΩM clusions ΩI 2 MULTIGRID AND2 MULTIGRID SADDLE-POINT AND2 MULTIGRID SADDLE-POINT PRECONDITIONERS AND2 MULTIGRID SADDLE-POINT PRECONDITIONERS AND SADDLE-POINT PRECONDITIONERS PRECONDITIONERS Figure 1: A domain with multiple inclusions and the unfitted meshes which encapsulates the matrix and inclusions. 3 FOR UNFITTED3 FOR FINITE UNFITTED ELEMENT3 FOR FINITE MODELLING UNFITTED ELEMENT3 FOR FINITE OF MODELLING UNFITTED ELEMENT FINITE OF MODELLING ELEMENT OF MODELLING OF 4 4 INCLUSIONS4 INCLUSIONS4 INCLUSIONS INCLUSIONS 1 2 n influence of these external forces, the domain under goes deformation, denoted as u := fuM ; uI ; uI ;:::; uI g. The deformation i or the displacement field is defined as a sufficiently regular function for the matrix domain as uM and the inclusions as uI . 5 On the interfaces, we assume5 the perfectHardik bonding Kothari, condition,5 Rolf such thatHardik Krause the displacement Kothari, field5 Rolf is definedHardik Krause to be Kothari, continuous. RolfHardik Krause Kothari, Rolf Krause The equation of the equilibrium, the boundary conditions and the interface conditions on such domain are written as 6 6 Affiliation6 Affiliation6 Affiliation Affiliation r · σ + f = 0 in Ω; 7 Institute of Computational7 Institute Science, of Universit`adella Computational7 SvizzeraInstitute Science, italiana, of Universit`adella Computational Lugano,7 Svizzera SwitzerlandInstitute Science, italiana, of Universit`adella Computational Lugano, Svizzera Switzerland Science, italiana, Universit`adella Lugano, SvizzeraSwitzerland italiana, Lugano, Switzerland u = 0 on @ΩD; 8 8 [email protected], [email protected] [email protected], [email protected] [email protected],(1) [email protected]@usi.ch, [email protected] σn = tN on @ΩN ; i ui = 0 on Γ ; for i = f1; 2; : : : ; ng; 9 Key words: Unfitted finite9 Key element words:J method,K Unfitted multigrid finite9 Key element method, words: method, saddle-pointUnfitted multigrid finite9 problemKey element method, words: method, saddle-pointUnfitted multigrid finite problem element method, method, saddle-point multigrid problem method, saddle-point problem where, σ denotes the Cauchy stress tensor and n denotes the outward normal. The jump of the displacement field is defined 10 Abstract. Ini this work,10 Abstract. we considerIn the this modeling work,10 Abstract. we of inclusionsconsiderIn thisthe in work,modeling the10 material weAbstract. consider of inclusions using theIn an modeling this in work, the material of we inclusions consider using in the the an modeling material of using inclusions an unfitted in the material using an unfitted as ui := uM jΓi − uI jΓi . We assume that the material is linear elastic and the constitutive law is provided by Hooke's law 11 σunfitted=J λtr(K ")I finite+ 2µ"; elementwhere λ and11 method.unfittedµ are Lam´eparameters, In finite the unfitted element and11 methods, method. tr(finite·) denotes element structuredIn the the trace unfittedmethod. operator. background11 methods, In Infinite the this meshes unfitted work, element structured we are considermethods, method. background the structured In the meshes unfitted background are methods, meshes structured are used background and meshes are used and 1 T 12 linearizedused and strain only tensor the underlying", given12 asused" finite:= and2 r element onlyu + (r theu) space underlying. 12 is modifiedonly finite the to elementunderlying incorporate space finite12 theis modified elementonly discontinuities, the spaceto underlying incorporate is modified finite the toelement discontinuities, incorporate space is the modified discontinuities, to incorporate such as the discontinuities, such as 13 such as inclusions. Hence,13 such the unfitted as inclusions. methods Hence, provide13 inclusions. the a unfitted more flexible Hence, methods framework the provide13 unfittedinclusions. fora more methods modeling flexible Hence, provide framework the unfitted a more for flexible methods modeling framework provide a for more modeling flexible the framework for modeling the 14 2.1the materials Unfitted with discretization multiple14 the inclusions. materials with We employ multiple14 materials the inclusions. method with of multiple We Lagrange employ14 inclusions.materials multipliers the method We with employ for of multiple Lagrange the inclusions. method multipliers of Lagrange We employ for multipliers the method for of enforcing Lagrange multipliers for enforcing 15 Weenforcing use a Cartesian the interface mesh Teh,15 conditions whichenforcing is assumed between the to interface bethe quasi-uniform inclusions15 conditionsthe and interface and shape between matrix, regular. conditions the which The inclusions15 mesh between givestheTeh interfaceis rise andthe fitted to matrix, inclusions to the conditions the domain which and between givesmatrix, rise the this to inclusions gives the rise and to the matrix, linear this system gives rise to the linear system i i 16 Ω,linear
Load more

Recommended publications
  • Multigrid Methods
    Multigrid Methods Volker John Winter Semester 2013/14 Contents 1 Literature 2 2 Model Problems 3 3 Detailed Investigation of Classical Iterative Schemes 8 3.1 General Aspects of Classical Iterative Schemes . .8 3.2 The Jacobi and Damped Jacobi Method . 10 3.3 The Gauss{Seidel Method and the SOR Method . 15 3.4 Summary . 18 4 Grid Transfer 19 4.1 Algorithms with Coarse Grid Systems, the Residual Equation . 19 4.2 Prolongation or Interpolation . 21 4.3 Restriction . 23 5 The Two-Level Method 27 5.1 The Coarse Grid Problem . 27 5.2 General Approach for Proving the Convergence of the Two-Level Method . 30 5.3 The Smoothing Property of the Damped Jacobi Iteration . 32 5.4 The Approximation Property . 33 5.5 Summary . 36 6 The Multigrid Method 37 6.1 Multigrid Cycles . 37 6.2 Convergence of the W-cycle . 39 6.3 Computational Work of the Multigrid γ-Cycle . 45 7 Algebraic Multigrid Methods 49 7.1 Components of an AMG and Definitions . 49 7.2 Algebraic Smoothness . 52 7.3 Coarsening . 56 7.4 Prolongation . 58 7.5 Concluding Remarks . 60 8 Outlook 61 1 Chapter 1 Literature Remark 1.1 Literature. There are several text books about multigrid methods, e.g., • Briggs et al. (2000), easy to read introduction, • Hackbusch (1985), the classical book, sometimes rather hard to read, • Shaidurov (1995), • Wesseling (1992), an introductionary book, • Trottenberg et al. (2001). 2 2 Chapter 2 Model Problems Remark 2.1 Motivation. The basic ideas and properties of multigrid methods will be explained in this course on two model problems.
    [Show full text]
  • Hybrid Particle-Grid Water Simulation Using Multigrid Pressure Solver Per Karlsson
    LiU-ITN-TEK-G--14/006-SE Hybrid Particle-Grid Water Simulation using Multigrid Pressure Solver Per Karlsson 2014-03-13 Department of Science and Technology Institutionen för teknik och naturvetenskap Linköping University Linköpings universitet nedewS ,gnipökrroN 47 106-ES 47 ,gnipökrroN nedewS 106 47 gnipökrroN LiU-ITN-TEK-G--14/006-SE Hybrid Particle-Grid Water Simulation using Multigrid Pressure Solver Examensarbete utfört i Medieteknik vid Tekniska högskolan vid Linköpings universitet Per Karlsson Handledare George Baravdish Examinator Camilla Forsell Norrköping 2014-03-13 Upphovsrätt Detta dokument hålls tillgängligt på Internet – eller dess framtida ersättare – under en längre tid från publiceringsdatum under förutsättning att inga extra- ordinära omständigheter uppstår. Tillgång till dokumentet innebär tillstånd för var och en att läsa, ladda ner, skriva ut enstaka kopior för enskilt bruk och att använda det oförändrat för ickekommersiell forskning och för undervisning. Överföring av upphovsrätten vid en senare tidpunkt kan inte upphäva detta tillstånd. All annan användning av dokumentet kräver upphovsmannens medgivande. För att garantera äktheten, säkerheten och tillgängligheten finns det lösningar av teknisk och administrativ art. Upphovsmannens ideella rätt innefattar rätt att bli nämnd som upphovsman i den omfattning som god sed kräver vid användning av dokumentet på ovan beskrivna sätt samt skydd mot att dokumentet ändras eller presenteras i sådan form eller i sådant sammanhang som är kränkande för upphovsmannens litterära eller konstnärliga anseende eller egenart. För ytterligare information om Linköping University Electronic Press se förlagets hemsida http://www.ep.liu.se/ Copyright The publishers will keep this document online on the Internet - or its possible replacement - for a considerable time from the date of publication barring exceptional circumstances.
    [Show full text]
  • Multigrid Algorithms for High Order Discontinuous Galerkin Methods
    Multigrid algorithms for high order discontinuous Galerkin methods Paola F. Antonietti1, Marco Sarti2, and Marco Verani3 Abstract I n this paper we study the performance of h- and p-multigrid algorithms for high order Discontinuous Galerkin discretizations of elliptic problems. We test the performance of the multigrid schemes employing a wide class of smoothers and considering both two- and three-dimensional test cases. 1 Introduction In the framework of multigrid solvers for Discontinuous Galerkin (DG) schemes, the first contributions are due to Gopalakrishnan and Kanschat [16] and Brenner and Zhao [10]. In [16] a V-cycle preconditioner for a Sym- metric Interior Penalty (SIP) discretization of an elliptic problem is analyzed. They prove that the condition number of the preconditioned system is uni- formly bounded with respect to the mesh size and the number of levels. In [10] V-cycle, F-cycle and W-cycle multigrid schemes for SIP discretizations are presented and analyzed, employing the additive theory developed in [8, 9]. A uniform bound for the error propagation operator is shown provided the number of smoothing steps is large enough. All the previously cited works focus on low order, i.e., linear, DG approximations. With regard to high order DG discretizations, h- and p-multigrid schemes are successfully em- ployed for the numerical solution of many different kinds of problems, see e.g. [14, 20, 22, 21, 24, 6], even if only few theoretical results are available that show the convergence properties of the underlying algorithms. In the MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano ITALY.
    [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]
  • Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method
    Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method James W. Lottes¤ and Paul F. Fischery February 4, 2004 Abstract We study the performance of the multigrid method applied to spectral element (SE) discretizations of the Poisson and Helmholtz equations. Smoothers based on finite element (FE) discretizations, overlapping Schwarz methods, and point-Jacobi are con- sidered in conjunction with conjugate gradient and GMRES acceleration techniques. It is found that Schwarz methods based on restrictions of the originating SE matrices converge faster than FE-based methods and that weighting the Schwarz matrices by the inverse of the diagonal counting matrix is essential to effective Schwarz smoothing. Sev- eral of the methods considered achieve convergence rates comparable to those attained by classic multigrid on regular grids. 1 Introduction The availability of fast elliptic solvers is essential to many areas of scientific computing. For unstructured discretizations in three dimensions, iterative solvers are generally optimal from both work and storage standpoints. Ideally, one would like to have computational complexity that scales as O(n) for an n-point grid problem in lRd, implying that the it- eration count should be bounded as the mesh is refined. Modern iterative methods such as multigrid and Schwarz-based domain decomposition achieve bounded iteration counts through the introduction of multiple representations of the solution (or the residual) that allow efficient elimination of the error at each scale. The theory for these methods is well established for classical finite difference (FD) and finite element (FE) discretizations, and order-independent convergence rates are often attained in practice. For spectral element (SE) methods, there has been significant work on the development of Schwarz-based methods that employ a combination of local subdomain solves and sparse global solves to precondition conjugate gradient iteration.
    [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]
  • Multigrid Method Based on a Space-Time Approach with Standard Coarsening for Parabolic Problems
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CWI's Institutional Repository Applied Mathematics and Computation 317 (2018) 25–34 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Multigrid method based on a space-time approach with standard coarsening for parabolic problems Sebastião Romero Franco a,b, Francisco José Gaspar c, Marcio Augusto Villela ∗ Pinto d, Carmen Rodrigo e, a Department of Mathematics, State University of Centro-Oeste, Irati, PR, Brazil b Graduate Program in Numerical Methods in Engineering, Federal University of Paraná, Curitiba, PR, Brazil c Centrum Wiskunde & Informatica (CWI), Science Park 123, P.O. Box 94079, Amsterdam 1090, The Netherlands d Department of Mechanical Engineering, Federal University of Paraná, Curitiba, PR, Brazil e IUMA and Applied Mathematics Department, University of Zaragoza, María de Luna, 3, Zaragoza 50018, Spain a r t i c l e i n f o a b s t r a c t MSC: In this work, a space-time multigrid method which uses standard coarsening in both tem- 00-01 poral and spatial domains and combines the use of different smoothers is proposed for the 99-00 solution of the heat equation in one and two space dimensions. In particular, an adaptive Keywords: smoothing strategy, based on the degree of anisotropy of the discrete operator on each Space-time multigrid grid-level, is the basis of the proposed multigrid algorithm. Local Fourier analysis is used Local Fourier analysis for the selection of the crucial parameter defining such an adaptive smoothing approach.
    [Show full text]
  • Electrostatic PIC with Adaptive Cartesian Mesh
    Electrostatic PIC with adaptive Cartesian mesh Vladimir Kolobov1,2 and Robert Arslanbekov1 1 CFD Research Corporation, Huntsville, AL, USA 2 The University of Alabama in Huntsville, Huntsville, AL 35899, USA E-mail: [email protected] Abstract. We describe an initial implementation of an electrostatic Particle-in-Cell (ES-PIC) module with adaptive Cartesian mesh in our Unified Flow Solver framework. Challenges of PIC method with cell-based adaptive mesh refinement (AMR) are related to a decrease of the particle-per-cell number in the refined cells with a corresponding increase of the numerical noise. The developed ES-PIC solver is validated for capacitively coupled plasma, its AMR capabilities are demonstrated for simulations of streamer development during high-pressure gas breakdown. It is shown that cell-based AMR provides a convenient particle management algorithm for exponential multiplications of electrons and ions in the ionization events. 1. Introduction Particle-in-Cell (PIC) is a mature technology widely used for plasma simulations. There are three versions of PIC: an electrostatic (EC) PIC uses Poisson equation for calculating electric field, an electromagnetic (EM) PIC employs full-wave Maxwell solver for calculations of electric and magnetic fields, and a quasi-static (QS) PIC uses a quasi-static version of the Maxwell equations (such as Darwin model) for calculation of the EM fields. Surprisingly, PIC solvers with adaptive mesh refinement (AMR) are relatively rare. It is worth mentioning the WARP code for accelerator applications [1], and an explicit EM-PIC code of Fujimoto [2] for astrophysics applications, both using block-structured AMR. In this paper, we describe an initial implementation of an ES-PIC module in our Unified Flow Solver (UFS) framework [3].
    [Show full text]
  • Hp-Multigrid As Smoother Algorithm for Higher Order Discontinuous Galerkin Discretizations of Advection Dominated Flows Part I
    HP-MULTIGRID AS SMOOTHER ALGORITHM FOR HIGHER ORDER DISCONTINUOUS GALERKIN DISCRETIZATIONS OF ADVECTION DOMINATED FLOWS PART I. MULTILEVEL ANALYSIS J.J.W. VAN DER VEGT∗ AND S. RHEBERGENy AMS subject classifications. Primary 65M55; Secondary 65M60, 76M10. Abstract. The hp-Multigrid as Smoother algorithm (hp-MGS) for the solution of higher order accurate space-(time) discontinuous Galerkin discretizations of advection dominated flows is pre- sented. This algorithm combines p-multigrid with h-multigrid at all p-levels, where the h-multigrid acts as smoother in the p-multigrid. The performance of the hp-MGS algorithm is further improved using semi-coarsening in combination with a new semi-implicit Runge-Kutta method as smoother. A detailed multilevel analysis of the hp-MGS algorithm is presented to obtain more insight into the theoretical performance of the algorithm. As model problem a fourth order accurate space-time dis- continuous Galerkin discretization of the advection-diffusion equation is considered. The multilevel analysis shows that the hp-MGS algorithm has excellent convergence rates, both for low and high cell Reynolds numbers and on highly stretched meshes. Key words. multigrid algorithms, discontinuous Galerkin methods, higher order accurate dis- cretizations, space-time methods, Runge-Kutta methods, Fourier analysis, multilevel analysis 1. Introduction. Discontinuous Galerkin finite element methods are well suited to obtain higher order accurate discretizations on unstructured meshes. The use of basis functions which are only weakly coupled to neighboring elements results in a local discretization which allows, in combination with hp-mesh adaptation, the efficient capturing of detailed structures in the solution, and is also beneficial for parallel computing.
    [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]