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