Multigrid algorithms for a vertex–centered covolume method for elliptic problems

Summary. We analyze V–cycle multigrid algorithms for a class of perturbed problems whose perturbation in the bilinear form preserves the convergence properties of the multigrid algorithm of the original problem. As an application, we study the convergence of multigrid algorithms for a covolume method or a vertex–centered finite volume element method for variable coefficient elliptic problems on polygonal domains. As in standard finite element methods, the V–cycle algorithm with one pre-smoothing converges with a rate independent of the number of levels. Various types of smoothers including point or line Jacobi, and Gauss-Seidel relaxation are considered. Received August 19, 1999 / Revised version received July 10, 2000 / Published online June 7, 2001     

Rate of Convergence for some constraint decomposition methods for nonlinear variational inequalities

Summary. Some general subspace correction algorithms are proposed for a convex optimization problem over a convex constraint subset. One of the nontrivial applications of the algorithms is the solving of some obstacle problems by multilevel domain decomposition and multigrid methods. For domain decomposition and multigrid methods, the rate of convergence for the algorithms for obstacle problems is of the same order as the rate of convergence for jump coefficient linear elliptic problems. In order to analyse the convergence rate, we need to decompose a finite element function into a sum of functions from the subspaces and also satisfying some constraints. A special nonlinear interpolation operator is introduced for decomposing the functions. Received December 13, 2001 / Revised version received February 19, 2002 / Published online June 17, 2002 This work was partially supported by the Norwegian Research Council under projects 128224/431 and SEP-115837/431.     

A cascadic multigrid algorithm for the Stokes equations

A variant of multigrid schemes for the Stokes problem is discussed. In particular, we propose and analyse a cascadic version for the Stokes problem. The analysis of the transfer between the grids requires special care in order to establish that the complexity is the same as that for classical multigrid algorithms. Received September 10, 1997 / Revised version received February 20, 1998     

Multigrid methods for a parameter dependent problem in primal variables

Summary. In this paper we consider multigrid methods for the parameter dependent problem of nearly incompressible materials. We construct and analyze multilevel-projection algorithms, which can be applied to the mixed as well as to the equivalent, non-conforming finite element scheme in primal variables. For proper norms, we prove that the smoothing property and the approximation property hold with constants that are independent of the small parameter. Thus we obtain robust and optimal convergence rates for the W-cycle and the variable V-cycle multigrid methods. The numerical results pretty well conform the robustness and optimality of the multigrid methods proposed. Received June 17, 1998 / Revised version received October 26, 1998 / Published online September 7, 1999     

Multigrid and multilevel methods for quadratic spline collocation

Multigrid methods are developed and analyzed for quadratic spline collocation equations arising from the discretization of one-dimensional second-order differential equations. The rate of convergence of the two-grid method integrated with a damped Richardson relaxation scheme as smoother is shown to be faster than 1/2, independently of the step-size. The additive multilevel versions of the algorithms are also analyzed. The development of quadratic spline collocation multigrid methods is extended to two-dimensional elliptic partial differential equations. Multigrid methods for quadratic spline collocation methods are not straightforward: because the basis functions used with quadratic spline collocation are not nodal basis functions, the design of efficient restriction and extension operators is nontrivial. Experimental results, with V-cycle and full multigrid, indicate that suitably chosen multigrid iteration is a very efficient solver for the quadratic spline collocation equations. Supported by Communications and Information Technology Ontario (CITO), Canada. Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

A characterization of mapping unstructured grids onto structured grids and using multigrid as a preconditioner

Many problems based on unstructured grids provide a natural multigrid framework due to using an adaptive gridding procedure. When the grids are saved, even starting from just a fine grid problem poses no serious theoretical difficulties in applying multigrid. A more difficult case occurs when a highly unstructured grid problem is to be solved with no hints how the grid was produced. Here, there may be no natural multigrid structure and applying such a solver may be quite difficult to do. Since unstructured grids play a vital role in scientific computing, many modifications have been proposed in order to apply a fast, robust multigrid solver. One suggested solution is to map the unstructured grid onto a structured grid and then apply multigrid to a sequence of structured grids as a preconditioner. In this paper, we derive both general upper and lower bounds on the condition number of this procedure in terms of computable grid parameters. We provide examples to illuminate when this preconditioner is a useful (e. g.,p orh-p formulated finite element problems on semi-structured grids) or should be avoided (e.g., typical computational fluid dynamics (CFD) or boundary layer problems). We show that unless great care is taken, this mapping can lead to a system with a high condition number which eliminates the advantage of the multigrid method. This work was partially supported by ONR Grant # N0014-91-J-1576.     

A multigrid method for a Petrov-Galerkin discretization of the Stokes equations

Summary We introduce a multigrid method for the solution of the discrete Stokes equations, arising from a Petrov-Galerkin formulation. The stiffness matrix is nonsymmetric but coercive, hence we consider smoothing iterations which are not suitable for usual indefinite problems. In this report, we prove convergence for a multigrid method with Richardson iteration in the smoothing part.     

Convergence analysis of a multigrid method for convection–diffusion equations

Summary. This paper is concerned with the convergence analysis of robust multigrid methods for convection-diffusion problems. We consider a finite difference discretization of a 2D model convection-diffusion problem with constant coefficients and Dirichlet boundary conditions. For the approximate solution of this discrete problem a multigrid method based on semicoarsening, matrix-dependent prolongation and restriction and line smoothers is applied. For a multigrid W-cycle we prove an upper bound for the contraction number in the euclidean norm which is smaller than one and independent of the mesh size and the diffusion/convection ratio. For the contraction number of a multigrid V-cycle a bound is proved which is uniform for a class of convection-dominated problems. The analysis is based on linear algebra arguments only. Received April 26, 2000 / Published online June 20, 2001     

Multigrid for the mortar element method for P1 nonconforming element

In this paper, a multigrid algorithm is presented for the mortar element method for P1 nonconforming element. Based on the theory developed by Bramble, Pasciak, Xu in [5], we prove that the W-cycle multigrid is optimal, i.e. the convergence rate is independent of the mesh size and mesh level. Meanwhile, a variable V-cycle multigrid preconditioner is constructed, which results in a preconditioned system with uniformly bounded condition number. Received May 11, 1999 / Revised version received April 1, 2000 / Published online October 16, 2000     

A conjugate gradient method and a multigrid algorithm for Morley s finite element approximation of the biharmonic equation

Summary The numerical solution of the linear equations arising from Morley's nonconforming displacement method is studied. A suitable preconditioning for the conjugate gradient method is described. Furthermore, the nonconformity of the discretization necessitates some non-standard ingredients of multigrid algorithms.     

Efficient geometric multigrid implementation for triangular grids

This paper deals with a stencil-based implementation of a geometric multigrid method on semi-structured triangular grids (triangulations obtained by regular refinement of an irregular coarse triangulation) for linear finite element methods. An efficient and elegant procedure to construct these stencils using a reference stencil associated to a canonical hexagon is proposed. Local Fourier Analysis (LFA) is applied to obtain asymptotic convergence estimates. Numerical experiments are presented to illustrate the efficiency of this geometric multigrid algorithm, which is based on a three-color smoother.     

Multilevel Schwarz methods

Summary We consider the solution of the algebraic system of equations which result from the discretization of second order elliptic equations. A class of multilevel algorithms are studied using the additive Schwarz framework. We establish that the condition number of the iteration operators are bounded independent of mesh sizes and the number of levels. This is an improvement on Dryja and Widlund's result on a multilevel additive Schwarz algorithm, as well as Bramble, Pasciak and Xu's result on the BPX algorithm. Some multiplicative variants of the multilevel methods are also considered. We establish that the energy norms of the corresponding iteration operators are bounded by a constant less than one, which is independent of the number of levels. For a proper ordering, the iteration operators correspond to the error propagation operators of certain V-cycle multigrid methods, using Gauss-Seidel and damped Jacobi methods as smoothers, respectively.This work was supported in part by the National Science Foundation under Grants NSF-CCR-8903003 at Courant Institute of Mathematical Sciences, New York University and NSF-ASC-8958544 at Department of Computer Science, University of Maryland.     

MULTIGRID ALGORITHM FOR THE COUPLING SYSTEM OF NATURAL BOUNDARY ELEMENT METHOD AND FINITE ELEMENT METHOD FOR UNBOUNDED DOMAIN PROBLEMS

In this paper, some V-cycle multigrid algorithms are presented for the coupling system arising from the discretization of the Dirichlet exterior problem by coupling the natural boundary element method and finite element method. The convergence of these multigrid algorithms is obtained even with only one smoothing on all levels. The rate of convergence is found uniformly bounded independent of the number of levels and the mesh sizes of all levels, which indicates that these multigrid algorithms are optimal. Some numerical results are also reported.     

Optimal Multigrid Methods with New Transfer Operators Based on Finite Difference Approximations

We constructed new interpolation operator in multigrid methods, which is efficient to transfer residual error from coarse grid to fine grid. This operator used idea of solving local residual equation using the standard stencil and the skewed stencil of the centered difference approximation to the Laplacian operator. We also compared our new multigrid methods with traditional multigrid methods, and found that new method is optimal.     

The cascadic multigrid method for elliptic problems

Summary. The paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods based on the cascade principle of [10]. From the multigrid point of view, no correction cycles are needed; from the cascade principle view, a basic iteration method without any preconditioner is used at successive refinement levels. For a prescribed error tolerance on the final level, more iterations must be spent on coarser grids in order to allow for less iterations on finer grids. A first candidate of such a cascadic multigrid method was the recently suggested cascadic conjugate gradient method of [9], in short CCG method, whichused the CG method as basic iteration method on each level. In [18] it has been proven, that the CCG method is accurate with optimal complexity for elliptic problems in 2D and quasi-uniform triangulations. The present paper simplifies that theory and extends it to more general basic iteration methods like the traditional multigrid smoothers. Moreover, an adaptive control strategy for the number of iterations on successive refinement levels for possibly highly non-uniform grids is worked out on the basis of a posteriori estimates. Numerical tests confirm the efficiency and robustness of the cascadic multigrid method. Received November 12, 1994 / Revised version received October 12, 1995     

Error estimates of the lumped mass finite element method for semilinear elliptic problems

We are concerned with the semilinear elliptic problems. We first investigate the L²-error estimate for the lumped mass finite element method. We then use the cascadic multigrid method to solve the corresponding discrete problem. On the basis of the finite element error estimates, we prove the optimality of the proposed multigrid method. We also report some numerical results to support the theory.     

MULTIGRID METHODS FOR OBSTACLE PROBLEMS

In this review,we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition,to projected subspace decomposition and truncated multigrid.In addition,we present a novel globally convergent inexact active set method which is closely related to truncated multigrid.The numerical properties of algorithms are carefully assessed by means of a degenerate problem and a problem with a complicated coincidence set.     

Steplength optimization and linear multigrid methods

Summary Recently steplength parameters have been used in linear multigrid methods. In this paper we give a theoretical analysis of the effects of steplength optimization in a rather general framework which covers two different implementations of steplength optimization in standard multigrid methods.     

Patch-adaptive multilevel iteration

Themultilevel adaptive iteration is an attempt to improve both the robustness and efficiency of iterative sparse system solvers. Unlike in most other iterative methods, the order of processing and sequence of operations is not determined a priori. The method consists of a relaxation scheme with an active set strategy and can be viewed as an efficient implementation of the Gauß-Southwell relaxation. With this strategy, computational work is focused on where it can efficiently improve the solution quality. To obtain full efficiency, the algorithm must be used on a multilevel structure. This algorithm is then closely related to multigrid or multilevel preconditioning algorithms, and can be shown to have asymptotically optimal convergence. In this paper the focus is on a variant that uses data structures with a locally uniform grid refinement. The resulting grid system consists of a collection of patches where each patch is a uniform rectangular grid and where adaptive refinement is accomplished by arranging the patches flexibly in space. This construction permits improved implementations that better exploit high performance computer designs. This will be demonstrated by numerical examples.     

High-order discretization and multigrid solution of elliptic nonlinear constrained optimal control problems

High(-mixed)-order finite difference discretization of optimality systems arising from elliptic nonlinear constrained optimal control problems are discussed. For the solution of these systems, an efficient and robust multigrid algorithm is presented. Theoretical and experimental results show the advantages of higher-order discretization and demonstrate that the proposed multigrid scheme is able to solve efficiently constrained optimal control problems also in the limit case of bang-bang control.