Multigrid Methods for Solving the Convection-Diffusion Equations Chin-Tien Wu National Center for Theoretical Sciences Mathematics Division Tsing-Hua University.

Slides:



Advertisements
Similar presentations
Stable Fluids A paper by Jos Stam.
Advertisements

Steady-state heat conduction on triangulated planar domain May, 2002
Optimal transport methods for mesh generation Chris Budd (Bath), JF Williams (SFU)
School of something FACULTY OF OTHER School of Computing An Adaptive Numerical Method for Multi- Scale Problems Arising in Phase-field Modelling Peter.
Mutigrid Methods for Solving Differential Equations Ferien Akademie 05 – Veselin Dikov.
05/11/2005 Carnegie Mellon School of Computer Science Aladdin Lamps 05 Combinatorial and algebraic tools for multigrid Yiannis Koutis Computer Science.
CSE 245: Computer Aided Circuit Simulation and Verification Matrix Computations: Iterative Methods (II) Chung-Kuan Cheng.
Chapter 8 Elliptic Equation.
Improvement of a multigrid solver for 3D EM diffusion Research proposal final thesis Applied Mathematics, specialism CSE (Computational Science and Engineering)
1 Iterative Solvers for Linear Systems of Equations Presented by: Kaveh Rahnema Supervisor: Dr. Stefan Zimmer
Computer Science & Engineering Department University of California, San Diego SPICE Diego A Transistor Level Full System Simulator Chung-Kuan Cheng May.
1 Numerical Solvers for BVPs By Dong Xu State Key Lab of CAD&CG, ZJU.
CS 290H 7 November Introduction to multigrid methods
Applied Linear Algebra - in honor of Hans SchneiderMay 25, 2010 A Look-Back Technique of Restart for the GMRES(m) Method Akira IMAKURA † Tomohiro SOGABE.
MULTISCALE COMPUTATIONAL METHODS Achi Brandt The Weizmann Institute of Science UCLA
Geometric (Classical) MultiGrid. Hierarchy of graphs Apply grids in all scales: 2x2, 4x4, …, n 1/2 xn 1/2 Coarsening Interpolate and relax Solve the large.
July 11, 2006 Comparison of Exact and Approximate Adjoint for Aerodynamic Shape Optimization ICCFD 4 July 10-14, 2006, Ghent Giampietro Carpentieri and.
Modern iterative methods For basic iterative methods, converge linearly Modern iterative methods, converge faster –Krylov subspace method Steepest descent.
Image Reconstruction Group 6 Zoran Golic. Overview Problem Multigrid-Algorithm Results Aspects worth mentioning.
A Bezier Based Approach to Unstructured Moving Meshes ALADDIN and Sangria Gary Miller David Cardoze Todd Phillips Noel Walkington Mark Olah Miklos Bergou.
Algebraic MultiGrid. Algebraic MultiGrid – AMG (Brandt 1982)  General structure  Choose a subset of variables: the C-points such that every variable.
Coupled Fluid-Structural Solver CFD incompressible flow solver has been coupled with a FEA code to analyze dynamic fluid-structure coupling phenomena CFD.
A posteriori Error Estimate - Adaptive method Consider the boundary value problem Weak form Discrete Equation Error bounds ( priori error )
Influence of (pointwise) Gauss-Seidel relaxation on the error Poisson equation, uniform grid Error of initial guess Error after 5 relaxation Error after.
1/36 Gridless Method for Solving Moving Boundary Problems Wang Hong Department of Mathematical Information Technology University of Jyväskyklä
Geometric (Classical) MultiGrid. Linear scalar elliptic PDE (Brandt ~1971)  1 dimension Poisson equation  Discretize the continuum x0x0 x1x1 x2x2 xixi.
Parallel Adaptive Mesh Refinement Combined With Multigrid for a Poisson Equation CRTI RD Project Review Meeting Canadian Meteorological Centre August.
1 Numerical Integration of Partial Differential Equations (PDEs)
© Fluent Inc. 9/5/2015L1 Fluids Review TRN Solution Methods.
Multigrid for Nonlinear Problems Ferien-Akademie 2005, Sarntal, Christoph Scheit FAS, Newton-MG, Multilevel Nonlinear Method.
Improving Coarsening and Interpolation for Algebraic Multigrid Jeff Butler Hans De Sterck Department of Applied Mathematics (In Collaboration with Ulrike.
Hybrid WENO-FD and RKDG Method for Hyperbolic Conservation Laws
Adaptive Multigrid FE Methods -- An optimal way to solve PDEs Zhiming Chen Institute of Computational Mathematics Chinese Academy of Sciences Beijing
Hans De Sterck Department of Applied Mathematics University of Colorado at Boulder Ulrike Meier Yang Center for Applied Scientific Computing Lawrence Livermore.
Introduction to Scientific Computing II From Gaussian Elimination to Multigrid – A Recapitulation Dr. Miriam Mehl.
ParCFD Parallel computation of pollutant dispersion in industrial sites Julien Montagnier Marc Buffat David Guibert.
The Finite Element Method A Practical Course
A particle-gridless hybrid methods for incompressible flows
Discontinuous Galerkin Methods Li, Yang FerienAkademie 2008.
1 Computational Methods II (Elliptic) Dr. Farzad Ismail School of Aerospace and Mechanical Engineering Universiti Sains Malaysia Nibong Tebal Pulau.
The swiss-carpet preconditioner: a simple parallel preconditioner of Dirichlet-Neumann type A. Quarteroni (Lausanne and Milan) M. Sala (Lausanne) A. Valli.
7. Introduction to the numerical integration of PDE. As an example, we consider the following PDE with one variable; Finite difference method is one of.
© Fluent Inc. 11/24/2015J1 Fluids Review TRN Overview of CFD Solution Methodologies.
CSE 245: Computer Aided Circuit Simulation and Verification Matrix Computations: Iterative Methods I Chung-Kuan Cheng.
Parallel Solution of the Poisson Problem Using MPI
Introduction to Scientific Computing II Multigrid Dr. Miriam Mehl Institut für Informatik Scientific Computing In Computer Science.
Introduction to Scientific Computing II Multigrid Dr. Miriam Mehl.
Lecture 21 MA471 Fall 03. Recall Jacobi Smoothing We recall that the relaxed Jacobi scheme: Smooths out the highest frequency modes fastest.
Introduction to Scientific Computing II
Discretization for PDEs Chunfang Chen,Danny Thorne Adam Zornes, Deng Li CS 521 Feb., 9,2006.
Generalized Finite Element Methods
1 Chapter 7 Numerical Methods for the Solution of Systems of Equations.
Partial Derivatives Example: Find If solution: Partial Derivatives Example: Find If solution: gradient grad(u) = gradient.
Consider Preconditioning – Basic Principles Basic Idea: is to use Krylov subspace method (CG, GMRES, MINRES …) on a modified system such as The matrix.
F. Fairag, H Tawfiq and M. Al-Shahrani Department of Math & Stat Department of Mathematics and Statistics, KFUPM. Nov 6, 2013 Preconditioning Technique.
A Parallel Hierarchical Solver for the Poisson Equation Seung Lee Deparment of Mechanical Engineering
Multipole-Based Preconditioners for Sparse Linear Systems. Ananth Grama Purdue University. Supported by the National Science Foundation.
Conjugate gradient iteration One matrix-vector multiplication per iteration Two vector dot products per iteration Four n-vectors of working storage x 0.
“Solve the equations right” Mathematical problem Validation :
Solving Linear Systems Ax=b
© Fluent Inc. 1/10/2018L1 Fluids Review TRN Solution Methods.
Convergence in Computational Science
CSE 245: Computer Aided Circuit Simulation and Verification
A Multigrid Tutorial part two
Introduction to Multigrid Method
Introduction to Scientific Computing II
Investigators Tony Johnson, T. V. Hromadka II and Steve Horton
Numerical Linear Algebra
Comparison of CFEM and DG methods
Numerical Modeling Ramaz Botchorishvili
Presentation transcript:

Multigrid Methods for Solving the Convection-Diffusion Equations Chin-Tien Wu National Center for Theoretical Sciences Mathematics Division Tsing-Hua University

Motivation 1. Incompressible Navier-Stokes equation models many flows in practices. Linearized backward Euler Discretization Oseen Problem Saddle Point Problem

2.A preconditioning technique proposed by Silvester, Elman, Kay and Wathen for solving linearized incompressible Naiver-Stokes has been shown to be very efficient. (J. Comput. and Appl. Math No 128 P ). 3. For achieving efficiency, a robust Poisson solver and a scalar convection-diffusion solver are needed. 4. Mulrigrid is efficient for solving Poisson problem. Can we have a robust multigrid solver for the convection-diffusion problems?

Convection-Diffusion Equation The problem we consider here is where for some constant c 0 and b, c and f are sufficiently smooth.

What are the foundations for building an efficient and accurate solver? I.Stable discretization methods for FEM. II.A good mesh. III.Reliable error estimation. IV.Fast iterative linear solver. Challenges in solving the convection-diffusion problem 1.When the convection is dominant,  <<1, the solution has sharp gradients due to the presents of Dirichlet outflow boundaries or discontinuity in inflow boundaries. 2.The associated linear system is not symmetric.

I. Discretization Galerkin FEM Streamline diffusion upwinding FEM by Hughes and Brooks 1979 Discontinuous Galerkin FEM by Reed and Hill 1973 Edge-averaged FEM by Xu and Zikatanov 1999 (the linear system form this discretization is an M-matrix)

Example 1: Characteristic and downstream layers Solution from SDFEM discretization on 32x32 grid Solution from Galerkin discretization on 32x32 grid

II. Methods for producing a good mesh? 1.Delaunay triangulation (DT): DT maximizes the minimal angle of the triangulation. 2.Advancing Front algorithm (AF): controls the element shape through control variables such as element stretch ratio (by Löhner 1988). 3. Advancing front Delaunay triangulation: combination of DT and AF (by Mavriplis 1993, Müller 1993 and Marcum 1995). 4. Longest side bisection: produces nested grids and guarantee the minimal angle on the fine grid is greater than or equal to half of the minimal angle on the coarse grid (by Rivara 1984). 5. Mesh relaxation includes moving mesh by equidistribution, MMPDE (by Huang, Ren and Russell 1994), and moving mesh FEM (by Carlson and Miller 1994, and Baines 1994).

III. A posteriori error indicators are used to pinpoint where the errors are large. A posteriori error estimation based on residual is proposed by Babuška and Rheinboldt 1978 A posteriori error estimation based on solving a local problem is proposed by Bank and Weiser 1985 A posteriori error estimation based on recovered gradient is proposed by Zienkiewicz and Zhu 1987 A posteriori error estimations for the convection-diffusion problems are proposed by Verfürth 1998, Kay and Silvester 2001.

Adaptive Solution Strategies Strategy I : In mesh refinement, we employee the maximum marking strategy: Solve  Compute error indicator  Refine mesh i ii i (Kay and Silvester 2001)

Improve Solution Accuracy by Mesh Refinement Adaptive meshes with threshold 0.01 in the maximum marking strategy Contour plots of solutions on adaptive meshes Problem 1:

IV. Solving Linear System Ax=b by Iterative Methods Methods: Stationary Methods: Jacobi, Gauss Seidel (GS), SOR. Krylov Subspace Methods: GMRES, by Saad and Schultz 1986, and MINRES, by Paige and Saunders Multigrid Methods: Geometric multigrid (MG), by Fedorenko 1961, and algebraic multigrid (AMG), by Ruge and Stüben Why Multigrid? Numerical Scheme Operations for Square/Cube 2-D 3-D Gaussian elimination O(N 2 ) O(N 7/3 ) Jacobi O(N 2 ) O(N 5/3 ) Conjugate Gradient O(N 3/2 ) O(N 4/3 ) Multigrid O(Nlog(N)) O(Nlog(N)) Multigrid convergence is independent with problem size.

Why multigrid works? 1. Relaxation methods converge slowly but smooth the error quickly. Consider Richardson relaxation: Fourier analysis shows that, after m relaxation and choosing 2. Smooth error modes are more oscillatory on coarse grids. Smooth errors can be better corrected by relaxation on coarser grids.

Multigrid (I) Multigrid (MG) Algorithm: MG Error reduction operator: Pre-smoothing onlyPost-smoothing only

Multigrid (II) Adaptive refinement MG V-cycle Restriction Prolongation AMG V-cycle AMG Coarsening Restriction Prolongation

Multigrid (GMG) and Algebraic Multigrid (AMG) MGAMG 1.A priori generated coarse grids are needed. Coarse grids need to be generated based on geometric information of the domain. 2.Interpolation operators are defined independent with coarsening process. 3.Smoother is not always fixed. 1.A priori generated coarse grids are not needed! Coarse grids are generated by algebraic coarsening from matrix on fine grid. 2. Interpolation operators are defined dynamically in coarsening process. 3.Smoother is fixed.

Multigrid Components Smoothing operator E s : Forward H-line GS Backward H-line GS Forward V-line GS Backward V-line GS : linear interpolation R/S coarsening AMG Discretization on V H GMG Coarse grid operatorRestriction Prolongation

GMG Convergence Smoothing property: Approximation property: Convergence needs robust smoothers together with semi-coarsening and operator dependent prolongation for the convection-diffusion equation (Reusken 2002). Choices of interploations: Linear interpolation: Operator-dependent interpolation: De Zeeuw 1990 Recent results:

GMG Convergence for Problem 1 Theorem 1: Horizontal line Gauss-Seidel (HGS) converges for Theorem 2.: Geometric multigrid with HGS smoother and bilinear interpolation converge and x x x mesh x x x mesh HGS convergence on rectangular mesh MG convergence on rectangular mesh Auxiliary inequalities:

AMG convergence Smoothing assumption: Approximation assumption: AMG works when A is a symmetric positive definite M-matrix.

Smooth error is characterized by Smoother errors vary slowly in the direction of strong connection, from e i to e j, where are large. AMG coarsening should be done in the direction of the strong connections. In the coarsening process, interpolation weights are computed so that the approximation assumption is satisfied. (detail see Ruge and Stüben 1985)

AMG works for Problem 1 Smooth error varies slowly along strong connected direction when h>>  2/3. R/S AMG coarsening and interpolation work. Smooth error e s satisfies Assuming the Galerkin coarse grid correction satisfied the approximation property, inequalities in the auxiliary lemmas and the approximation assumption imply AMG converges more rapidly than GMG as long as

AMG Coarsening Algorithm (I)

AMG Coarsening Algorithm (II)

AMG Coarsening AMG coarsening with strong connection parameter  /h <<  << 0.25 C-point SDFEM discretization of Problem 1 with has matrix stencil:

AMG Coarsening on Uniform Meshes Problem 1: b=(0,1) Flow field Coarse grids from AMG coarseningCoarse grids from GMG coarsening

Problem 2: circulating flow b=(2y(1-x 2 ),2x(1-y 2 )) Coarse grids from AMG coarsening Flow fieldCoarse grids from GMG coarsening

GMG and AMG as a Solver (I) On the uniform mesh: On the adaptive mesh: Problem 1: AMGGMGLevel AMGGMGLevel AMGGMGLevel  =  =  =10 -4  =  =10 -3  = AMGGMGlevel AMGGMGlevel AMGGMGlevel

GMG and AMG as a Solver (II) Problem 2: On the uniform mesh: On the adaptive mesh:

GMG and AMG as a Preconditioner of GMRES Problem 1: Problem 2: 1288GMRES-AMG GMRES-GMG GMRES-HGS GMRES 1188AMG GMG  1484GMRES-AMG 36165GMRES-GMG GMRES-HGS GMRES 1486AMG 59229GMG  Iterative steps on uniform mesh Iterative steps on adaptive mesh GMRES-AMG GMRES-GMG GMRES-HGS ---GMRES AMG GMG  16 10GMRES-AMG 14128GMRES-GMG GMRES-HGS ---GMRES AMG 13198GMG  Iterative steps on uniform mesh Iterative steps on adaptive mesh

Stopping Criterion for Iterative Solver u : the weak solution of the convection-diffusion equation, u h : the finite element solution u h,n : the approximate iterative solution of u h. Given a prescribed tolerance . If || u-u h || < 0.5 , clearly, we only have to ask || u h,n - u h || < 0.5  for large enough n. From a posteriori estimation || u-u h || < c , where c is some constant, it is natural to acquire the iterative solution satisfies a stopping tolerance such that C1. || u h,n - u h || < c , For adaptive mesh refinement, it is also desirable that C2.    n (  n is the error indicator computed from u h,n ) Question : 1.What stopping criteria should be imposed for iterative solutions? 2.What solver requires least iterative steps to satisfy the stopping criteria?

Basic Ideas on Deriving the Stopping Criteria

With the assumptions we prove, when Kay and Silvester’s error indicator is used for mesh refniement. Stopping criterion for satisfying C1.

With the assumption we prove, when Kay and Silvester’s error indicator is used for mesh refinement. Stopping criterion for satisfying C2.

Verification of assumption of Theorem Kay and Silvester’s indicator: Problem 1: Characteristic and downstream layers (ii) Number of points in refined meshes

Problem 2: Flow with Closed Characteristics (ii) Kay and Silvester’s indicator: Verification of assumption of Theorem Number of points in refined meshes

Problem 1: Characteristic and downstream layers (i) Verfürth’s indicator: Verification of assumption of Theorem Number of points in refined meshes

Problem 2: Flow with Closed Characteristics (i) Verfürth’s indicator: Verification of assumption of Theorem Number of points in refined meshes

Recent Numerical Results of Multigrid Methods on Solving Convection-Diffusion Equations GMRES and BiCGSTAB accelerated W-cycle MG with alternative zebra line GS smoother and upwind prolongation achieves h-independent convergence on problems with closed characteristics in which upwind discretization is used (by Oosterlee and Washio 1998). BICGSTAB accelerated V- and F- cycle AMG with symmetric GS smoother shows very slightly h-dependent convergence for problems with closed characteristics in which upwind discretization is used (Trottenberg, Oosterlee and Schüller: Multigrid p ). GMRES accelerated V-cycle MG with line GS smoother and bilinear, upwind or matrix- dependent prolongation achieves h-independent convergence for the model problems in which SDFEM discretization is used (by Ramage 1999). By GMRES acceleration, improvement on convergence of MG and AMG is obtained on both uniform and adaptive meshes. GMRES accelerated AMG is an attractive black-box solver for the SDFEM discretized convection-diffusion equation.

Thank You

Conclusion 1.SDFEM discretization is more stable than Galerkin discretization. 2.Our error-adapted mesh refinement is able to produce a good mesh for resolving the boundary layers. 3.On adaptive meshes, MG is a robust solver only for problems with only characteristics and AMG is robust for problems with only exponential layers. Both MG and AMG are good preconditioner for GMRES. 4.Fewer iterative steps are required for the MG solver to satisfy our stopping criteria ( in Theorem and Theorem 5.2.6) than to satisfy the heuristic tolerance (residual less than ). No such saving can be seen if GMRES is used. The total saving of computation works is significant (can be more than half of the total works with heuristic tolerance).

Furture works 1.Investigate the performance of different linear solvers from EAFEM. 2.Deriving a posteriori error estimations for EAFEM. 3.Numerical studies on the a posteriori error estimation by Kunert on anisotropic mesh generated by error-adapted refinement process. 4.Extend our stopping criteria to different problems. 5.Solve more difficult problems such as Navier-Stokes equations by more accurate and efficient methods.

Discretization : Solutions from Galerkin and SDFEM discretizations. Error estimator : Reliability of a posteriori error estimators, based on residual and based on solving a local problem. Mesh improvement: Moving mesh and error-adapted mesh refinement strategy Linear Solver : 1. Introduction of geometric multigrid and algebraic multigrid 2.Convergence of line Gauss-Seidel and multigrid with line Gauss-Seidel smoother when 3.Comparison of geometric multigrid and algebraic multigrid methods as a solver and as a preconditioner of GMRES. 4.Stopping criteria for iterative solvers based on a posteriori error bounds. In this talk, we will discuss:

Galerking and SDFEM Galerkin method : SDFEM method :

Problem 1: Downstream boundary layers: Consider is a solution of the convection-diffusion equation: Dirichlet boundary condition given by where with Solution from Galerkin discretization on 32x32 grid Solution from SDFEM discretization on 32x32 grid Mesh for compuitng error

A Posteriori Error Estimation Based on Residual Proposed by Verfürth Upper Bound: Local lower Bound: where and

A Posteriori Error Estimation Base on Solving a Local Problem Proposed by Kay and Silvester Upper Bound: Local lower Bound:

Comparison of VR and KS Error Indicators (i) Problem 1:

Comparison of VR and KS Error Indicators (ii) / / / /64 64x6432x3216x168x8  / / / /64 64x6432x3216x168x8  E  of VR indicator E  of KS indicator Comparison of the global effectivity indices / / / /64 64x6432x3216x168x8  / / / /64 64x6432x3216x168x8  E T of VR indicator E T of KS indicator Comparison of the local effectivity indices

Moving Mesh and Error-Adapted Mesh Refinement Why mesh movement? A heuristic strategy for increasing the accuracy of numerical solution. Mesh movement tends to cluster nodes in the area with sharp gradient. We move meshes by following the equidistribution principle where Kay and Silvester’s error indicator is employed as a monitor function. Why error-adapted mesh refinement? To resolve boundary layers, regular mesh refinement may take too many steps and generate too many nodes. Error-adapted mesh refinement is able to cluster nodes to the locations where sharp gradients appear. Moving mesh destroy the nested grid structure from adaptive refinement process. Error-adapted refinement maintain nested grid structure.

Mesh Moving by Equidistribution

Soluiton on refinemnet mesh Soluiton on movement + refinement mesh Moving Mesh (I) Problem 2

Moving Mesh (II) Solution on refinement mesh Solution on movement + refinement mesh Problem 3 : IAHR/CEGB:

Error-Adapted Mesh Refinement 1.Compute recovered error indicator for every node 2.Compute external force for each edge 3.Modify external force to and 4.

Choice of error sensitivity parameter  Problem 2:

Error-Adapted Mesh Refinement v.s. Regular Mesh Refinement (I) Problem 2 with ε=10 -4 Number of node Regular Refinement Error-adapted Refinement refinement steps are performed. 2 Threshold  = 0.5 in maximum marking strategy.

Variant IAHR/CEGB problem Flow Field:

Error-Adapted Mesh Refinement v.s. Regular Mesh Refinement (II) variant IAHR/CEGB problem with ε=10 -4 Number of node Regular Refinement Error-adapted Refinement refinement steps are performed. 2 Threshold  = 0.25 in maximum marking strategy.

Driven Cavity Flow for Re=100

GMRES

FEM Discretization