Multigrid finite element methods on semi-structured triangular grids
|
|
- Brenda Johnston
- 5 years ago
- Views:
Transcription
1 XXI Congreso de Ecuaciones Diferenciales y Aplicaciones XI Congreso de Matemática Aplicada Ciudad Real, -5 septiembre 009 (pp. 8) Multigrid finite element methods on semi-structured triangular grids F.J. Gaspar, J.L. Gracia, F.J. Lisbona, C. Rodrigo Applied Mathematics Department, University of Zaragoza, Pedro Cerbuna, Zaragoza, Spain. s: fjgaspar@unizar.es, jlgracia@unizar.es, lisbona@unizar.es, carmenr@unizar.es. Keywords: Geometric multigrid, Fourier analysis, triangular grids Abstract We are interested in the design of efficient geometric multigrid methods on hierarchical triangular grids for problems in two dimensions. Fourier analysis is a well-known useful tool in multigrid for the prediction of two-grid convergence rates which has been used mainly for rectangular grids. This analysis can be extended straightforwardly to triangular grids by using an appropriate expression of the Fourier transform in a new coordinate systems, both in space and frequency variables. With the help of the Fourier Analysis, efficient geometric multigrid methods for the Laplace problem on hierarchical triangular grids are designed. Numerical results show that the Local Fourier Analysis (LFA) predicts with high accuracy the multigrid convergence rates for different geometries.. Introduction Multigrid methods [3, 7, 8] are among the most efficient numerical algorithms for solving the large algebraic linear equation systems arising from discretizations of partial differential equations. In geometric multigrid, a hierarchy of grids must be proposed. For an irregular domain, it is very common to apply a refinement process to an unstructured input grid, such as Bank s algorithm, used in the codes PLTMG [] and KASKADE [5], obtaining a particular hierarchy of globally unstructured grids suitable for use with geometric multigrid. A simpler approach to generating the nested grids consists in carrying out several steps of repeated regular refinement, for example by dividing each triangle into four congruent triangles [4]. An important step in the analysis of PDE problems using finite element methods (FEM) is the construction of the large sparse matrix corresponding to the system of equations to be solved. For discretizations of problems defined on structured grids with constant
2 F.J. Gaspar, J.L. Gracia, F.J. Lisbona, C. Rodrigo coefficients, explicit assembly of the global matrix for the finite element method is not necessary, and the discrete operator can be implemented using stencil-based operations. For the previously described hierarchical grid, one stencil suffices to represent the discrete operator at nodes inside a triangle of the coarsest grid, and standard assembly process is only used on the coarsest grid. Therefore, this technique is used in this paper since it can be very efficient and is not subject to the same memory limitations as unstructured grid representation. LFA (also called local mode analysis []) is a powerful tool for the quantitative analysis and design of efficient multigrid methods for general problems on rectangular grids. Recently, a generalization to structured triangular grids, which is based on an expression of the Fourier transform in new coordinate systems in space and frequency variables, has been proposed in [6]. In that paper some smoothers (Jacobi, Gauss Seidel, three color and block line) have been analyzed and compared by LFA, the three color smoother turning out to be the best choice for almost equilateral triangles. The organization of the paper is as follows. In Section the way in which Fourier analysis can be extended to multigrid methods for discretizations on regular non-rectangular grids is explained. The proposed relaxation methods are introduced in Section 3. Some Fourier analysis results for Poisson problem are presented in Section 4 in order to justify the choice of the smoothers, and finally, in Section 5 a numerical experiment is performed to show the efficiency of the proposed algorithm.. Fourier analysis on non-orthogonal grids This analysis is based on the multi-dimensional Fourier transform using coordinates in an orthonormal basis of R. We aim for discretizations on triangular grids in the twodimensional case, so the key to accomplishing this is to introduce the two-dimensional Fourier transform using coordinates in non-orthogonal bases fitting the new structure of the grid. Let {e,e } be a unitary basis of R, 0 < γ < π the angle between the vectors of the basis and {e,e } its reciprocal basis, i.e., e i e j = δ ij, i,j, where δ ij is Kronecker s delta. If {e,e } is the canonical basis, we will denote by y = (y,y ), y = (y,y ) and y = (y,y ) the coordinates of a point in the bases {e,e }, {e,e } and {e,e }, respectively. By applying variable changes x = F(x ) and θ = G(θ ) to the usual Fourier transform formula, the Fourier transform with coordinates in a non-orthogonal basis results in û(g(θ )) = sin γ e ig(θ ) F(x ) u(f(x )) dx. π R In a similar way, the back transformation formula is given by u(f(x )) = e ig(θ ) F(x )û(g(θ )) dθ. π sin γ R Since the new bases are reciprocal bases, the inner product G(θ ) F(x ) is given by θ x + θ x. Now using the previous expressions, a discrete Fourier transform for nonrectangular grids can be introduced. Let h = (h,h ) be a grid spacing and G h = {x = (x,x ) x i = k ih i, k i Z, i =,} a uniform infinite grid oriented in the directions e and e.
3 Multigrid finite element methods on semi-structured triangular grids Now, for a grid function u h, the discrete Fourier transform can be defined by û h (θ ) = h h sin γ π x G h e i(θ x +θ x ) u h (x ), where θ = (θ,θ ) Θ h = ( π/h,π/h ] ( π/h,π/h ] are the coordinates of the point θ e + θ e in the frequency space. Its back Fourier transformation is given by u h (x ) = e i(θ x +θ x ) û h (θ )dθ. () π sin γ Θ h From (), each discrete function u h (x ) with x G h, can be written as a formal linear combination of the discrete exponential functions ϕ h (θ,x ) = e iθ x e iθ x, called Fourier modes, which give rise to the Fourier space, F(G h ) = span{ϕ h (θ, ) θ Θ h }. Due to the fact that the grid and the frequency space are referred to as reciprocal bases, the Fourier modes have a formal expression, in terms of θ and x, similar to those in Cartesian coordinates. Therefore, the Local Fourier analysis on non rectangular grids can be performed straightforwardly. Figure : A regular triangular grid on a fixed coarse triangle T and its extension to an infinite grid. Let T h be a regular triangular grid on a fixed coarse triangle T ; see left picture of Figure. T h is extended to the infinite grid given before, where e and e are unit vectors indicating the direction of two of the edges of T, and such that T h = G h T, see right picture of Figure. Neglecting boundary conditions and/or connections with other neighboring triangles on the coarsest grid, the discrete problem L h u h = f h can be extended to the whole grid G h. It is straightforward to see that the grid functions ϕ h (θ,x ) are formal eigenfunctions of any discrete operator L h which can be given in stencil notation. Using standard coarsening, (H = (H,H ) = (h,h )), high and low frequency components on G h are distinguished in the way that the subset of low frequencies is Θ H = ( π/h,π/h ] ( π/h,π/h ], and the subset of high frequencies is Θ h \ Θ H. From these definitions, LFA smoothing and two-grid analysis can be performed as in rectangular grids, and smoothing factors for the relaxing methods µ, and two-grid convergence factors ρ, which give the asymptotic convergence behavior of the method, can be well defined. 3
4 F.J. Gaspar, J.L. Gracia, F.J. Lisbona, C. Rodrigo 3. Relaxing methods It is well known that multicolor relaxation procedures are very efficient smoothers for multigrid methods. Besides, they are well suited for parallel computation. Nevertheless, they have the disadvantage that the Fourier components of L h are not eigenfunctions of the relaxation operator, what makes their analysis more complicated. Three-color smoother and some line-wise smoothers are proposed as relaxing methods. These smoothers appear as a natural extension to triangular grids of some smoothers widely used on rectangular grids, as red-black Gauss-Seidel and line-wise relaxations of zebra type. To apply three-color smoother, the grid associated with a fixed refinement level η of a triangle T of the coarsest triangulation, G = {x = (x,x ) x j = k jh j, k j Z, j =,, k = 0,..., η, k = 0,...,k }, () is split into three disjoint subgrids, G i = {x = (x,x ) G x j = k jh j, j =,, k + k = i(mod 3)}, i = 0,,, each of them associated with a different color, as shown in Figure a), so that the unknowns of the same color have no direct connection with each other. The complete three-color smoothing operator is given by the product of three partial operators, S h = Sh S h S0 h. In each partial relaxation step, only the grid points of G i are processed, whereas the remaining points are not treated. For triangular grids, three different zebra smoothers can be defined on a triangle. We will denote them as zebra-red, zebra-black and zebra-green smoothers, since they correspond to each of the vertices of the triangle which can be associated with these colors. In Figure b) the zebra-type smoother corresponding to the red vertex is shown. They consist of two half steps. In the first half-step, odd lines parallel to the edges of the triangle are processed, whereas even lines are relaxed in the second step, in which the updated approximations on the odd lines are used. RED POINTS R BLACK POINTS GREEN POINTS B G a) b) Figure : a) Three-color smoother. b) Zebra-red smoother: approximations at points marked by are updated in the first half-step of the relaxation, those marked by in the second. In order to perform these smoothers, a splitting of the grid G into two different subsets G even and Godd is necessary. For each of the zebra smoothers these subgrids are 4
5 Multigrid finite element methods on semi-structured triangular grids defined in a different way, and the corresponding distinction between them is specified in Table, where k, and k are the indices of the grid points given in (). Thus, these three Relaxation Zebra-red k even k odd Zebra-black k even k odd Zebra-green k + k even k + k odd G even Table : Characterization of subgrids G even G odd and Godd for different zebra smoothers. smoothers Sh zr, SzB h and Sh zg are defined by the product of two partial operators. For example, if zebra-red smoother is considered, Sh zr = S zr even h S zr odd h where S zr even h is in charge of relaxing the points in G even and SzR odd h is responsible for the points in G odd. These smoothers are preferred to the lexicographic line-wise Gauss-Seidel because in spite of having the same computational cost, their smoothing factors are better than those of lexicographic line-wise relaxations as we will see further on. 4. Fourier analysis results for Laplace operator To analyze the influence of grid-geometry on the properties of multigrid methods, in this section we apply the LFA to discretizations of the Laplace operator by linear finite elements on a regular triangulation of a general triangle. The geometric parameters that we consider are two angles of the triangle, denoted here by α and β and the distance h between them. The components of the coarse grid correction are the standard coarsening, the natural coarse grid discretization, and as transfer operators we consider linear interpolation and its adjoint (I H h = 4 (Ih H) ) as the restriction. In Tables, 3 and 4 we show the smoothing factors µ ν +ν and the two grid convergence factors ρ for triangles with angles α = β = 60 0, α = 90 0, β = 45 0 and α = β = 75 0 respectively, and for different pre smoothing (ν ) and post smoothing (ν ) steps. We also display in these tables the experimentally measured W cycle convergence factors, ρ h, using nine levels of refinement, obtained with a right hand side zero and a random initial guess to avoid round-off errors. We have chosen W cycles to verify two grid convergence factors since they run at similar rates to the two grid rates but at less cost. In the case of Jacobi relaxation, we have used the optimal parameters w = /(5 ) for equilateral triangles, which can be analytically calculated, and the well known w = 0,8 for the five point discretization of the Laplace operator on rectangular grids. For the last triangle, we have performed an LFA showing that the optimal value is w =. From these tables, we can observe that the convergence factors are very well predicted by LFA in all cases. These convergence factors are improved when an equilateral triangle is considered, and we point out the exceptional convergence factors in Table associated with the three-color smoother. On the other hand, the highly satisfactory factors obtained for equilateral triangles worsen in Table 4,where an isosceles triangle with common angle 75 0 is considered, showing that none of these point-wise smoothers is robust over all angles. It can be seen that the 5
6 F.J. Gaspar, J.L. Gracia, F.J. Lisbona, C. Rodrigo Damped Jacobi Gauss-Seidel Three-colors ν,ν µ ν +ν ρ ρ h µ ν +ν ρ ρ h µ ν +ν ρ ρ h, , , Table : LFA results and measured W cycle convergence rates ρ h for equilateral triangles. Damped Jacobi Gauss-Seidel Three-colors ν,ν µ ν +ν ρ ρ h µ ν +ν ρ ρ h µ ν +ν ρ ρ h, , , Table 3: LFA results and measured W cycle convergence rates ρ h for α = 90 0, β = Jacobi Gauss-Seidel Three-colors ν,ν µ ν +ν ρ ρ h µ ν +ν ρ ρ h µ ν +ν ρ ρ h, , , Table 4: LFA results and measured W cycle convergence rates ρ h for α = β =
7 Multigrid finite element methods on semi-structured triangular grids Equilateral Isosceles (75 o ) Isosceles (85 o ) ν,ν µ ν +ν ρ µ ν +ν ρ µ ν +ν ρ, , , Table 5: LFA results obtained with zebra-type smoothers for different triangles. optimal values correspond to the case of almost equilateral triangles and the convergence factors deteriorates when any of the three angles becomes smaller. To overcome this difficulty we have designed three block line (or coupled) Gauss Seidel smoothers of zebra-type, introduced in Section 3. For almost equilateral triangles the smoothing and two grid factors for the point-wise Gauss Seidel and the zebra-type smoothers are quite similar, and we have obtained different results for non equilateral triangles. Therefore, these smoother perform very well for small values of its corresponding angle. Then, for each acute triangle we can always activate a zebra type smoother to have an optimal two grid convergence factor. This behavior can be seen in Table Numerical experiment We consider the model problem u = f. The right-hand side and the Dirichlet boundary conditions are such that the exact solution is u(x, y) = sin(πx) sin(πy). This problem is solved in an A-shaped domain, as it is shown in Figure 3a), and the coarsest mesh is composed of sixty triangles with different geometries, which are also depicted in the same figure. Nested meshes are constructed by regular refinement and the grid resulting after refining each triangle twice is shown in Figure 3b). Y Y Y Three-colorsmoother Block-line smoother X X X a) b) c) Figure 3: a) Computational domain and coarsest grid, b) Hierarchical grid obtained after two refinement levels, c) Different smoothers used in each triangle of the coarsest grid The considered problem has been discretized with linear finite elements, and the corresponding algebraic linear system has been solved with the geometric multigrid method proposed in previous sections, that is, we choose the more suitable smoother for each 7
8 F.J. Gaspar, J.L. Gracia, F.J. Lisbona, C. Rodrigo triangle of the coarsest triangulation, as we can see in Figure 3c), depending on the results predicted by LFA. From the local convergence factors predicted by LFA on each triangle, a global convergence factor of 0,44 is predicted by taking into account the worst of them. In order to see the robustness of the multigrid method with respect to the space discretization parameter h, in Figure 4 we show the convergence obtained, with an F(,)- cycle and the multigrid method proposed, for different numbers of refinement levels. The initial guess is taken as u(x, y) = and the stopping criterion is chosen as the maximum residual to be less than 0 6. Besides an asymptotic convergence factor about 0,43 has been obtained with a right hand side zero and a random initial guess. Note that Fourier two grid analysis predicts the convergence factors with a high degree of accuracy. e+008 e levels 7 levels 8 levels 9 levels 0 levels 0000 maximum residual e-006 e cycles Figure 4: Multigrid convergence F(,)-cycle for the model problem. Acknowledgements This research was partially supported by the project MEC/FEDER MTM and the Diputación General de Aragón. References [] R. Bank, PLTMG: A software package for solving elliptic partial differential equations. Users Guide Version 0.0, Department of Mathematics, University of California, 007. [] A. Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comput., 3 (977), [3] A. Brandt, Multigrid techniques: 984 guide with applications to fluid dynamics, GMD-Studie Nr. 85, Sankt Augustin, Germany, 984. [4] B. Bergen, T. Gradl, F. Hülsemann, U. Ruede, A massively parallel multigrid method for finite elements, Comput. Sci. Eng., 8 (006), [5] P. Deuflhard, P. Leinenand, H. Yserentant, Concepts of an adaptive hierarchical finite element code, Impact Comput. Sci. Engrg., (989), [6] F.J. Gaspar, J.L. Gracia and F.J. Lisbona, Fourier analysis for multigrid methods on triangular grids, SIAM J. Sci. Comput., 3 (009), [7] W. Hackbusch, Multi-grid methods and applications, Springer, Berlin, 985. [8] U. Trottenberg, C.W. Oosterlee, A. Schüller. Multigrid Academic Press, New York, 00. 8
Aspects of Multigrid
Aspects of Multigrid Kees Oosterlee 1,2 1 Delft University of Technology, Delft. 2 CWI, Center for Mathematics and Computer Science, Amsterdam, SIAM Chapter Workshop Day, May 30th 2018 C.W.Oosterlee (CWI)
More informationUniversity of Illinois at Urbana-Champaign. Multigrid (MG) methods are used to approximate solutions to elliptic partial differential
Title: Multigrid Methods Name: Luke Olson 1 Affil./Addr.: Department of Computer Science University of Illinois at Urbana-Champaign Urbana, IL 61801 email: lukeo@illinois.edu url: http://www.cs.uiuc.edu/homes/lukeo/
More informationKasetsart University Workshop. Multigrid methods: An introduction
Kasetsart University Workshop Multigrid methods: An introduction Dr. Anand Pardhanani Mathematics Department Earlham College Richmond, Indiana USA pardhan@earlham.edu A copy of these slides is available
More informationMultigrid Methods and their application in CFD
Multigrid Methods and their application in CFD Michael Wurst TU München 16.06.2009 1 Multigrid Methods Definition Multigrid (MG) methods in numerical analysis are a group of algorithms for solving differential
More informationc 2014 Society for Industrial and Applied Mathematics
SIAM J. SCI. COMPUT. Vol. 36, No. 3, pp. A1187 A1206 c 2014 Society for Industrial and Applied Mathematics A SIMPLE AND EFFICIENT SEGREGATED SMOOTHER FOR THE DISCRETE STOKES EQUATIONS FRANCISCO J. GASPAR,
More information1. Fast Iterative Solvers of SLE
1. Fast Iterative Solvers of crucial drawback of solvers discussed so far: they become slower if we discretize more accurate! now: look for possible remedies relaxation: explicit application of the multigrid
More informationFinite Element Multigrid Framework for Mimetic Finite Difference Discretizations
Finite Element Multigrid Framework for Mimetic Finite ifference iscretizations Xiaozhe Hu Tufts University Polytopal Element Methods in Mathematics and Engineering, October 26-28, 2015 Joint work with:
More informationNumerical Solution I
Numerical Solution I Stationary Flow R. Kornhuber (FU Berlin) Summerschool Modelling of mass and energy transport in porous media with practical applications October 8-12, 2018 Schedule Classical Solutions
More informationComparison of V-cycle Multigrid Method for Cell-centered Finite Difference on Triangular Meshes
Comparison of V-cycle Multigrid Method for Cell-centered Finite Difference on Triangular Meshes Do Y. Kwak, 1 JunS.Lee 1 Department of Mathematics, KAIST, Taejon 305-701, Korea Department of Mathematics,
More informationTHE EFFECT OF MULTIGRID PARAMETERS IN A 3D HEAT DIFFUSION EQUATION
Int. J. of Applied Mechanics and Engineering, 2018, vol.23, No.1, pp.213-221 DOI: 10.1515/ijame-2018-0012 Brief note THE EFFECT OF MULTIGRID PARAMETERS IN A 3D HEAT DIFFUSION EQUATION F. DE OLIVEIRA *
More informationEfficient implementation of box-relaxation multigrid methods for the poroelasticity problem on semi-structured grids
Monografías de la Real Academia de Ciencias de Zaragoza 33, 21 38, (2010. Efficient implementation of box-relaxation multigrid methods for the poroelasticity problem on semi-structured grids Francisco
More informationGeometric Multigrid Methods
Geometric Multigrid Methods Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University IMA Tutorial: Fast Solution Techniques November 28, 2010 Ideas
More informationINTRODUCTION TO MULTIGRID METHODS
INTRODUCTION TO MULTIGRID METHODS LONG CHEN 1. ALGEBRAIC EQUATION OF TWO POINT BOUNDARY VALUE PROBLEM We consider the discretization of Poisson equation in one dimension: (1) u = f, x (0, 1) u(0) = u(1)
More informationResearch Article Evaluation of the Capability of the Multigrid Method in Speeding Up the Convergence of Iterative Methods
International Scholarly Research Network ISRN Computational Mathematics Volume 212, Article ID 172687, 5 pages doi:1.542/212/172687 Research Article Evaluation of the Capability of the Multigrid Method
More informationEfficient smoothers for all-at-once multigrid methods for Poisson and Stokes control problems
Efficient smoothers for all-at-once multigrid methods for Poisson and Stoes control problems Stefan Taacs stefan.taacs@numa.uni-linz.ac.at, WWW home page: http://www.numa.uni-linz.ac.at/~stefant/j3362/
More informationINTERGRID OPERATORS FOR THE CELL CENTERED FINITE DIFFERENCE MULTIGRID ALGORITHM ON RECTANGULAR GRIDS. 1. Introduction
Trends in Mathematics Information Center for Mathematical Sciences Volume 9 Number 2 December 2006 Pages 0 INTERGRID OPERATORS FOR THE CELL CENTERED FINITE DIFFERENCE MULTIGRID ALGORITHM ON RECTANGULAR
More informationMultigrid absolute value preconditioning
Multigrid absolute value preconditioning Eugene Vecharynski 1 Andrew Knyazev 2 (speaker) 1 Department of Computer Science and Engineering University of Minnesota 2 Department of Mathematical and Statistical
More informationDiscrete Projection Methods for Incompressible Fluid Flow Problems and Application to a Fluid-Structure Interaction
Discrete Projection Methods for Incompressible Fluid Flow Problems and Application to a Fluid-Structure Interaction Problem Jörg-M. Sautter Mathematisches Institut, Universität Düsseldorf, Germany, sautter@am.uni-duesseldorf.de
More informationAlgebraic Multigrid as Solvers and as Preconditioner
Ò Algebraic Multigrid as Solvers and as Preconditioner Domenico Lahaye domenico.lahaye@cs.kuleuven.ac.be http://www.cs.kuleuven.ac.be/ domenico/ Department of Computer Science Katholieke Universiteit Leuven
More informationGeometric Multigrid Methods for the Helmholtz equations
Geometric Multigrid Methods for the Helmholtz equations Ira Livshits Ball State University RICAM, Linz, 4, 6 November 20 Ira Livshits (BSU) - November 20, Linz / 83 Multigrid Methods Aim: To understand
More informationCompact High Order Finite Difference Stencils for Elliptic Variable Coefficient and Interface Problems
Compact High Order Finite Difference Stencils for Elliptic Variable Coefficient and Interface Problems Daniel Ritter 1, Ulrich Rüde 1, Björn Gmeiner 1, Rochus Schmid 2 Copper Mountain, March 18th, 2013
More informationThe Effect of the Schedule on the CPU Time for 2D Poisson Equation
Trabalho apresentado no XXXV CNMAC, Natal-RN, 2014. The Effect of the Schedule on the CPU Time for 2D Poisson Equation Fabiane de Oliveira, State University of Ponta Grossa, Department of Mathematics and
More informationNew Multigrid Solver Advances in TOPS
New Multigrid Solver Advances in TOPS R D Falgout 1, J Brannick 2, M Brezina 2, T Manteuffel 2 and S McCormick 2 1 Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, P.O.
More informationBoundary value problems on triangular domains and MKSOR methods
Applied and Computational Mathematics 2014; 3(3): 90-99 Published online June 30 2014 (http://www.sciencepublishinggroup.com/j/acm) doi: 10.1164/j.acm.20140303.14 Boundary value problems on triangular
More informationJournal of Computational and Applied Mathematics. Multigrid method for solving convection-diffusion problems with dominant convection
Journal of Computational and Applied Mathematics 226 (2009) 77 83 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
More informationComputational Linear Algebra
Computational Linear Algebra PD Dr. rer. nat. habil. Ralf-Peter Mundani Computation in Engineering / BGU Scientific Computing in Computer Science / INF Winter Term 2018/19 Part 4: Iterative Methods PD
More informationarxiv: v2 [math.na] 2 Apr 2018
USING HIERARCHICAL MATRICES IN THE SOLUTION OF THE TIME-FRACTIONAL HEAT EQUATION BY MULTIGRID WAVEFORM RELAXATION XIAOZHE HU, CARMEN RODRIGO, AND FRANCISCO J. GASPAR arxiv:76.7632v2 [math.na] 2 Apr 28
More informationEFFICIENT MULTIGRID BASED SOLVERS FOR ISOGEOMETRIC ANALYSIS
6th European Conference on Computational Mechanics (ECCM 6) 7th European Conference on Computational Fluid Dynamics (ECFD 7) 1115 June 2018, Glasgow, UK EFFICIENT MULTIGRID BASED SOLVERS FOR ISOGEOMETRIC
More informationStabilization and Acceleration of Algebraic Multigrid Method
Stabilization and Acceleration of Algebraic Multigrid Method Recursive Projection Algorithm A. Jemcov J.P. Maruszewski Fluent Inc. October 24, 2006 Outline 1 Need for Algorithm Stabilization and Acceleration
More informationMULTIGRID METHODS FOR NONLINEAR PROBLEMS: AN OVERVIEW
MULTIGRID METHODS FOR NONLINEAR PROBLEMS: AN OVERVIEW VAN EMDEN HENSON CENTER FOR APPLIED SCIENTIFIC COMPUTING LAWRENCE LIVERMORE NATIONAL LABORATORY Abstract Since their early application to elliptic
More informationConstrained Minimization and Multigrid
Constrained Minimization and Multigrid C. Gräser (FU Berlin), R. Kornhuber (FU Berlin), and O. Sander (FU Berlin) Workshop on PDE Constrained Optimization Hamburg, March 27-29, 2008 Matheon Outline Successive
More informationSolving PDEs with Multigrid Methods p.1
Solving PDEs with Multigrid Methods Scott MacLachlan maclachl@colorado.edu Department of Applied Mathematics, University of Colorado at Boulder Solving PDEs with Multigrid Methods p.1 Support and Collaboration
More informationNumerical Analysis of the Double Porosity Consolidation Model
XXI Congreso de Ecuaciones Diferenciales y Aplicaciones XI Congreso de Matemática Aplicada Ciudad Real 21-25 septiembre 2009 (pp. 1 8) Numerical Analysis of te Double Porosity Consolidation Model N. Boal
More informationAlgorithms for Scientific Computing
Algorithms for Scientific Computing Finite Element Methods Michael Bader Technical University of Munich Summer 2016 Part I Looking Back: Discrete Models for Heat Transfer and the Poisson Equation Modelling
More informationarxiv: v1 [math.na] 6 Nov 2017
Efficient boundary corrected Strang splitting Lukas Einkemmer Martina Moccaldi Alexander Ostermann arxiv:1711.02193v1 [math.na] 6 Nov 2017 Version of November 6, 2017 Abstract Strang splitting is a well
More informationarxiv: v1 [math.na] 11 Jul 2011
Multigrid Preconditioner for Nonconforming Discretization of Elliptic Problems with Jump Coefficients arxiv:07.260v [math.na] Jul 20 Blanca Ayuso De Dios, Michael Holst 2, Yunrong Zhu 2, and Ludmil Zikatanov
More informationChapter Two: Numerical Methods for Elliptic PDEs. 1 Finite Difference Methods for Elliptic PDEs
Chapter Two: Numerical Methods for Elliptic PDEs Finite Difference Methods for Elliptic PDEs.. Finite difference scheme. We consider a simple example u := subject to Dirichlet boundary conditions ( ) u
More informationSolving Symmetric Indefinite Systems with Symmetric Positive Definite Preconditioners
Solving Symmetric Indefinite Systems with Symmetric Positive Definite Preconditioners Eugene Vecharynski 1 Andrew Knyazev 2 1 Department of Computer Science and Engineering University of Minnesota 2 Department
More informationarxiv: v1 [math.na] 15 Mar 2015
A Finite Element Framework for Some Mimetic Finite Difference Discretizations arxiv:1503.04423v1 [math.na] 15 Mar 2015 C. Rodrigo a,, F.J. Gaspar a, X. Hu b, L. Zikatanov c,d a Applied Mathematics Department
More informationIterative Methods and Multigrid
Iterative Methods and Multigrid Part 1: Introduction to Multigrid 2000 Eric de Sturler 1 12/02/09 MG01.prz Basic Iterative Methods (1) Nonlinear equation: f(x) = 0 Rewrite as x = F(x), and iterate x i+1
More informationAdaptive Time Space Discretization for Combustion Problems
Konrad-Zuse-Zentrum fu r Informationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Germany JENS LANG 1,BODO ERDMANN,RAINER ROITZSCH Adaptive Time Space Discretization for Combustion Problems 1 Talk
More informationMultigrid Acceleration of the Horn-Schunck Algorithm for the Optical Flow Problem
Multigrid Acceleration of the Horn-Schunck Algorithm for the Optical Flow Problem El Mostafa Kalmoun kalmoun@cs.fau.de Ulrich Ruede ruede@cs.fau.de Institute of Computer Science 10 Friedrich Alexander
More informationMathematics Research Report No. MRR 003{96, HIGH RESOLUTION POTENTIAL FLOW METHODS IN OIL EXPLORATION Stephen Roberts 1 and Stephan Matthai 2 3rd Febr
HIGH RESOLUTION POTENTIAL FLOW METHODS IN OIL EXPLORATION Stephen Roberts and Stephan Matthai Mathematics Research Report No. MRR 003{96, Mathematics Research Report No. MRR 003{96, HIGH RESOLUTION POTENTIAL
More informationIntroduction to Scientific Computing II Multigrid
Introduction to Scientific Computing II Multigrid Miriam Mehl Slide 5: Relaxation Methods Properties convergence depends on method clear, see exercises and 3), frequency of the error remember eigenvectors
More informationA greedy strategy for coarse-grid selection
A greedy strategy for coarse-grid selection S. MacLachlan Yousef Saad August 3, 2006 Abstract Efficient solution of the very large linear systems that arise in numerical modelling of real-world applications
More informationScientific Computing I
Scientific Computing I Module 8: An Introduction to Finite Element Methods Tobias Neckel Winter 2013/2014 Module 8: An Introduction to Finite Element Methods, Winter 2013/2014 1 Part I: Introduction to
More informationElliptic Problems / Multigrid. PHY 604: Computational Methods for Physics and Astrophysics II
Elliptic Problems / Multigrid Summary of Hyperbolic PDEs We looked at a simple linear and a nonlinear scalar hyperbolic PDE There is a speed associated with the change of the solution Explicit methods
More informationLecture 9 Approximations of Laplace s Equation, Finite Element Method. Mathématiques appliquées (MATH0504-1) B. Dewals, C.
Lecture 9 Approximations of Laplace s Equation, Finite Element Method Mathématiques appliquées (MATH54-1) B. Dewals, C. Geuzaine V1.2 23/11/218 1 Learning objectives of this lecture Apply the finite difference
More informationLehrstuhl Informatik V. Lehrstuhl Informatik V. 1. solve weak form of PDE to reduce regularity properties. Lehrstuhl Informatik V
Part I: Introduction to Finite Element Methods Scientific Computing I Module 8: An Introduction to Finite Element Methods Tobias Necel Winter 4/5 The Model Problem FEM Main Ingredients Wea Forms and Wea
More informationRobust solution of Poisson-like problems with aggregation-based AMG
Robust solution of Poisson-like problems with aggregation-based AMG Yvan Notay Université Libre de Bruxelles Service de Métrologie Nucléaire Paris, January 26, 215 Supported by the Belgian FNRS http://homepages.ulb.ac.be/
More information3D Space Charge Routines: The Software Package MOEVE and FFT Compared
3D Space Charge Routines: The Software Package MOEVE and FFT Compared Gisela Pöplau DESY, Hamburg, December 4, 2007 Overview Algorithms for 3D space charge calculations Properties of FFT and iterative
More informationAn Introduction of Multigrid Methods for Large-Scale Computation
An Introduction of Multigrid Methods for Large-Scale Computation Chin-Tien Wu National Center for Theoretical Sciences National Tsing-Hua University 01/4/005 How Large the Real Simulations Are? Large-scale
More informationChapter 5. Methods for Solving Elliptic Equations
Chapter 5. Methods for Solving Elliptic Equations References: Tannehill et al Section 4.3. Fulton et al (1986 MWR). Recommended reading: Chapter 7, Numerical Methods for Engineering Application. J. H.
More informationAdaptive algebraic multigrid methods in lattice computations
Adaptive algebraic multigrid methods in lattice computations Karsten Kahl Bergische Universität Wuppertal January 8, 2009 Acknowledgements Matthias Bolten, University of Wuppertal Achi Brandt, Weizmann
More informationSpectral element agglomerate AMGe
Spectral element agglomerate AMGe T. Chartier 1, R. Falgout 2, V. E. Henson 2, J. E. Jones 4, T. A. Manteuffel 3, S. F. McCormick 3, J. W. Ruge 3, and P. S. Vassilevski 2 1 Department of Mathematics, Davidson
More informationMultigrid Method for 2D Helmholtz Equation using Higher Order Finite Difference Scheme Accelerated by Krylov Subspace
201, TextRoad Publication ISSN: 2090-27 Journal of Applied Environmental and Biological Sciences www.textroad.com Multigrid Method for 2D Helmholtz Equation using Higher Order Finite Difference Scheme
More informationIntroduction to Multigrid Methods
Introduction to Multigrid Methods Chapter 9: Multigrid Methodology and Applications Gustaf Söderlind Numerical Analysis, Lund University Textbooks: A Multigrid Tutorial, by William L Briggs. SIAM 1988
More informationA fast method for the solution of the Helmholtz equation
A fast method for the solution of the Helmholtz equation Eldad Haber and Scott MacLachlan September 15, 2010 Abstract In this paper, we consider the numerical solution of the Helmholtz equation, arising
More informationDELFT UNIVERSITY OF TECHNOLOGY
DELFT UNIVERSITY OF TECHNOLOGY REPORT 06-09 Multigrid for High Dimensional Elliptic Partial Differential Equations on Non-equidistant Grids H. bin Zubair, C.W. Oosterlee, R. Wienands ISSN 389-6520 Reports
More informationAMG for a Peta-scale Navier Stokes Code
AMG for a Peta-scale Navier Stokes Code James Lottes Argonne National Laboratory October 18, 2007 The Challenge Develop an AMG iterative method to solve Poisson 2 u = f discretized on highly irregular
More informationMultigrid Solution of the Debye-Hückel Equation
Rochester Institute of Technology RIT Scholar Works Theses Thesis/Dissertation Collections 5-17-2013 Multigrid Solution of the Debye-Hückel Equation Benjamin Quanah Parker Follow this and additional works
More informationScientific Computing II
Technische Universität München ST 008 Institut für Informatik Dr. Miriam Mehl Scientific Computing II Final Exam, July, 008 Iterative Solvers (3 pts + 4 extra pts, 60 min) a) Steepest Descent and Conjugate
More informationMultigrid solvers for equations arising in implicit MHD simulations
Multigrid solvers for equations arising in implicit MHD simulations smoothing Finest Grid Mark F. Adams Department of Applied Physics & Applied Mathematics Columbia University Ravi Samtaney PPPL Achi Brandt
More informationFinite Element Methods
Solving Operator Equations Via Minimization We start with several definitions. Definition. Let V be an inner product space. A linear operator L: D V V is said to be positive definite if v, Lv > for every
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 11 Partial Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002.
More informationPoisson Equation in 2D
A Parallel Strategy Department of Mathematics and Statistics McMaster University March 31, 2010 Outline Introduction 1 Introduction Motivation Discretization Iterative Methods 2 Additive Schwarz Method
More informationPartial Differential Equations
Partial Differential Equations Introduction Deng Li Discretization Methods Chunfang Chen, Danny Thorne, Adam Zornes CS521 Feb.,7, 2006 What do You Stand For? A PDE is a Partial Differential Equation This
More informationFRIEDRICH-ALEXANDER-UNIVERSITÄT ERLANGEN-NÜRNBERG. Lehrstuhl für Informatik 10 (Systemsimulation)
FRIEDRICH-ALEXANDER-UNIVERSITÄT ERLANGEN-NÜRNBERG INSTITUT FÜR INFORMATIK (MATHEMATISCHE MASCHINEN UND DATENVERARBEITUNG) Lehrstuhl für Informatik 10 (Systemsimulation) Multigrid HowTo: A simple Multigrid
More informationRobust multigrid methods for nonsmooth coecient elliptic linear systems
Journal of Computational and Applied Mathematics 123 (2000) 323 352 www.elsevier.nl/locate/cam Robust multigrid methods for nonsmooth coecient elliptic linear systems Tony F. Chan a; ; 1, W.L. Wan b; 2
More informationANALYSIS AND COMPARISON OF GEOMETRIC AND ALGEBRAIC MULTIGRID FOR CONVECTION-DIFFUSION EQUATIONS
ANALYSIS AND COMPARISON OF GEOMETRIC AND ALGEBRAIC MULTIGRID FOR CONVECTION-DIFFUSION EQUATIONS CHIN-TIEN WU AND HOWARD C. ELMAN Abstract. The discrete convection-diffusion equations obtained from streamline
More informationRemarks on the analysis of finite element methods on a Shishkin mesh: are Scott-Zhang interpolants applicable?
Remarks on the analysis of finite element methods on a Shishkin mesh: are Scott-Zhang interpolants applicable? Thomas Apel, Hans-G. Roos 22.7.2008 Abstract In the first part of the paper we discuss minimal
More informationNumerical Programming I (for CSE)
Technische Universität München WT 1/13 Fakultät für Mathematik Prof. Dr. M. Mehl B. Gatzhammer January 1, 13 Numerical Programming I (for CSE) Tutorial 1: Iterative Methods 1) Relaxation Methods a) Let
More informationINTRODUCTION TO FINITE ELEMENT METHODS
INTRODUCTION TO FINITE ELEMENT METHODS LONG CHEN Finite element methods are based on the variational formulation of partial differential equations which only need to compute the gradient of a function.
More informationAn Accurate Fourier-Spectral Solver for Variable Coefficient Elliptic Equations
An Accurate Fourier-Spectral Solver for Variable Coefficient Elliptic Equations Moshe Israeli Computer Science Department, Technion-Israel Institute of Technology, Technion city, Haifa 32000, ISRAEL Alexander
More informationIn this chapter we study elliptical PDEs. That is, PDEs of the form. 2 u = lots,
Chapter 8 Elliptic PDEs In this chapter we study elliptical PDEs. That is, PDEs of the form 2 u = lots, where lots means lower-order terms (u x, u y,..., u, f). Here are some ways to think about the physical
More informationSimulating Solid Tumor Growth Using Multigrid Algorithms
Simulating Solid Tumor Growth Using Multigrid Algorithms Asia Wyatt Applied Mathematics, Statistics, and Scientific Computation Program Advisor: Doron Levy Department of Mathematics/CSCAMM Abstract In
More informationarxiv: v1 [math.na] 3 Nov 2018
arxiv:1811.1264v1 [math.na] 3 Nov 218 Monolithic mixed-dimensional multigrid methods for single-phase flow in fractured porous media Andrés Arrarás Francisco J. Gaspar Laura Portero Carmen Rodrigo November
More informationA PRECONDITIONER FOR THE HELMHOLTZ EQUATION WITH PERFECTLY MATCHED LAYER
European Conference on Computational Fluid Dynamics ECCOMAS CFD 2006 P. Wesseling, E. Oñate and J. Périaux (Eds) c TU Delft, The Netherlands, 2006 A PRECONDITIONER FOR THE HELMHOLTZ EQUATION WITH PERFECTLY
More informationA MULTIGRID-BASED SHIFTED-LAPLACIAN PRECONDITIONER FOR A FOURTH-ORDER HELMHOLTZ DISCRETIZATION.
A MULTIGRID-BASED SHIFTED-LAPLACIAN PRECONDITIONER FOR A FOURTH-ORDER HELMHOLTZ DISCRETIZATION. N.UMETANI, S.P.MACLACHLAN, C.W. OOSTERLEE Abstract. In this paper, an iterative solution method for a fourth-order
More informationMultilevel Preconditioning of Graph-Laplacians: Polynomial Approximation of the Pivot Blocks Inverses
Multilevel Preconditioning of Graph-Laplacians: Polynomial Approximation of the Pivot Blocks Inverses P. Boyanova 1, I. Georgiev 34, S. Margenov, L. Zikatanov 5 1 Uppsala University, Box 337, 751 05 Uppsala,
More informationA Hybrid Method for the Wave Equation. beilina
A Hybrid Method for the Wave Equation http://www.math.unibas.ch/ beilina 1 The mathematical model The model problem is the wave equation 2 u t 2 = (a 2 u) + f, x Ω R 3, t > 0, (1) u(x, 0) = 0, x Ω, (2)
More informationEfficient numerical solution of the Biot poroelasticity system in multilayered domains
Efficient numerical solution of the Biot poroelasticity system Anna Naumovich Oleg Iliev Francisco Gaspar Fraunhofer Institute for Industrial Mathematics Kaiserslautern, Germany, Spain Workshop on Model
More informationMultigrid and Domain Decomposition Methods for Electrostatics Problems
Multigrid and Domain Decomposition Methods for Electrostatics Problems Michael Holst and Faisal Saied Abstract. We consider multigrid and domain decomposition methods for the numerical solution of electrostatics
More informationNUMERICAL ALGORITHMS FOR A SECOND ORDER ELLIPTIC BVP
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI (S.N. MATEMATICĂ, Tomul LIII, 2007, f.1 NUMERICAL ALGORITHMS FOR A SECOND ORDER ELLIPTIC BVP BY GINA DURA and RĂZVAN ŞTEFĂNESCU Abstract. The aim
More informationA SHORT NOTE COMPARING MULTIGRID AND DOMAIN DECOMPOSITION FOR PROTEIN MODELING EQUATIONS
A SHORT NOTE COMPARING MULTIGRID AND DOMAIN DECOMPOSITION FOR PROTEIN MODELING EQUATIONS MICHAEL HOLST AND FAISAL SAIED Abstract. We consider multigrid and domain decomposition methods for the numerical
More informationBlock-Structured Adaptive Mesh Refinement
Block-Structured Adaptive Mesh Refinement Lecture 2 Incompressible Navier-Stokes Equations Fractional Step Scheme 1-D AMR for classical PDE s hyperbolic elliptic parabolic Accuracy considerations Bell
More informationBasic Aspects of Discretization
Basic Aspects of Discretization Solution Methods Singularity Methods Panel method and VLM Simple, very powerful, can be used on PC Nonlinear flow effects were excluded Direct numerical Methods (Field Methods)
More informationSOLVING MESH EIGENPROBLEMS WITH MULTIGRID EFFICIENCY
SOLVING MESH EIGENPROBLEMS WITH MULTIGRID EFFICIENCY KLAUS NEYMEYR ABSTRACT. Multigrid techniques can successfully be applied to mesh eigenvalue problems for elliptic differential operators. They allow
More informationPreface to the Second Edition. Preface to the First Edition
n page v Preface to the Second Edition Preface to the First Edition xiii xvii 1 Background in Linear Algebra 1 1.1 Matrices................................. 1 1.2 Square Matrices and Eigenvalues....................
More informationMultigrid Method ZHONG-CI SHI. Institute of Computational Mathematics Chinese Academy of Sciences, Beijing, China. Joint work: Xuejun Xu
Multigrid Method ZHONG-CI SHI Institute of Computational Mathematics Chinese Academy of Sciences, Beijing, China Joint work: Xuejun Xu Outline Introduction Standard Cascadic Multigrid method(cmg) Economical
More informationOn Multigrid for Phase Field
On Multigrid for Phase Field Carsten Gräser (FU Berlin), Ralf Kornhuber (FU Berlin), Rolf Krause (Uni Bonn), and Vanessa Styles (University of Sussex) Interphase 04 Rome, September, 13-16, 2004 Synopsis
More informationc 2004 Society for Industrial and Applied Mathematics
SIAM J. SCI. COMPUT. Vol. 25, No. 6, pp. 982 23 c 24 Society for Industrial and Applied Mathematics A BOUNDARY CONDITION CAPTURING MULTIGRID APPROAC TO IRREGULAR BOUNDARY PROBLEMS JUSTIN W. L. WAN AND
More informationME Computational Fluid Mechanics Lecture 5
ME - 733 Computational Fluid Mechanics Lecture 5 Dr./ Ahmed Nagib Elmekawy Dec. 20, 2018 Elliptic PDEs: Finite Difference Formulation Using central difference formulation, the so called five-point formula
More informationBootstrap AMG. Kailai Xu. July 12, Stanford University
Bootstrap AMG Kailai Xu Stanford University July 12, 2017 AMG Components A general AMG algorithm consists of the following components. A hierarchy of levels. A smoother. A prolongation. A restriction.
More informationOptimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 1/36
Optimal multilevel preconditioning of strongly anisotropic problems. Part II: non-conforming FEM. Svetozar Margenov margenov@parallel.bas.bg Institute for Parallel Processing, Bulgarian Academy of Sciences,
More informationSolving the stochastic steady-state diffusion problem using multigrid
IMA Journal of Numerical Analysis (2007) 27, 675 688 doi:10.1093/imanum/drm006 Advance Access publication on April 9, 2007 Solving the stochastic steady-state diffusion problem using multigrid HOWARD ELMAN
More informationHigh order, finite volume method, flux conservation, finite element method
FLUX-CONSERVING FINITE ELEMENT METHODS SHANGYOU ZHANG, ZHIMIN ZHANG, AND QINGSONG ZOU Abstract. We analyze the flux conservation property of the finite element method. It is shown that the finite element
More informationFLEXIBLE MULTIPLE SEMICOARSENING FOR THREE-DIMENSIONAL SINGULARLY PERTURBED PROBLEMS
SIAM J. SCI. COMPUT. c 1998 Society for Industrial and Applied Mathematics Vol. 19, No. 5, pp. 1646 1666, September 1998 014 FLEXIBLE MULTIPLE SEMICOARSENING FOR THREE-DIMENSIONAL SINGULARLY PERTURBED
More informationSUPERCONVERGENCE PROPERTIES FOR OPTIMAL CONTROL PROBLEMS DISCRETIZED BY PIECEWISE LINEAR AND DISCONTINUOUS FUNCTIONS
SUPERCONVERGENCE PROPERTIES FOR OPTIMAL CONTROL PROBLEMS DISCRETIZED BY PIECEWISE LINEAR AND DISCONTINUOUS FUNCTIONS A. RÖSCH AND R. SIMON Abstract. An optimal control problem for an elliptic equation
More informationNumerical Solution Techniques in Mechanical and Aerospace Engineering
Numerical Solution Techniques in Mechanical and Aerospace Engineering Chunlei Liang LECTURE 3 Solvers of linear algebraic equations 3.1. Outline of Lecture Finite-difference method for a 2D elliptic PDE
More information