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     

The hierarchical basis multigrid method for convection-diffusion equations

Summary We make a theoretical study of the application of a standard hierarchical basis multigrid iteration to the convection diffusion equation, discretized using an upwind finite element discretizations. We show behavior that in some respects is similar to the symmetric positive definite case, but in other respects is markedly different. In particular, we find the rate of convergence depends significantly on parameters which measure the strength of the upwinding, and the size of the convection term. Numerical calculations illustrating some of these effects are given.The work of this author was supported by the Office of Naval Research under contract N00014-89J-1440.The work of this author was supported by the Office of Naval Research under contract N00014-89J-1440.     

A Dual-Primal FETI method for incompressible Stokes equations

In this paper, a dual-primal FETI method is developed for incompressible Stokes equations approximated by mixed finite elements with discontinuous pressures. The domain of the problem is decomposed into nonoverlapping subdomains, and the continuity of the velocity across the subdomain interface is enforced by introducing Lagrange multipliers. By a Schur complement procedure, the solution of an indefinite Stokes problem is reduced to solving a symmetric positive definite problem for the dual variables, i.e., the Lagrange multipliers. This dual problem is solved by the conjugate gradient method with a Dirichlet preconditioner. In each iteration step, both subdomain problems and a coarse level problem are solved by a direct method. It is proved that the condition number of this preconditioned dual problem is independent of the number of subdomains and bounded from above by the square of the product of the inverse of the inf-sup constant of the discrete problem and the logarithm of the number of unknowns in the individual subdomains. Numerical experiments demonstrate the scalability of this new method. This work is based on a doctoral dissertation completed at Courant Institute of Mathematical Sciences, New York University. This work was supported in part by the National Science Foundation under Grants NSF-CCR-9732208, and in part by the U.S. Department of Energy under contract DE-FG02-92ER25127.     

Fourier analysis of a robust multigrid method for convection-diffusion equations

Summary. We consider a two-grid method for solving 2D convection-diffusion problems. The coarse grid correction is based on approximation of the Schur complement. As a preconditioner of the Schur complement we use the exact Schur complement of modified fine grid equations. We assume constant coefficients and periodic boundary conditions and apply Fourier analysis. We prove an upper bound for the spectral radius of the two-grid iteration matrix that is smaller than one and independent of the mesh size, the convection/diffusion ratio and the flow direction; i.e. we have a (strong) robustness result. Numerical results illustrating the robustness of the corresponding multigrid -cycle are given. Received October 14, 1994     

Schwarz preconditioners for the spectral element discretization of the steady Stokes and Navier-Stokes equations

Summary. The - spectral element discretization of the Stokes equation gives rise to an ill-conditioned, indefinite, symmetric linear system for the velocity and pressure degrees of freedom. We propose a domain decomposition method which involves the solution of a low-order global, and several local problems, related to the vertices, edges, and interiors of the subdomains. The original system is reduced to a symmetric equation for the velocity, which can be solved with the conjugate gradient method. We prove that the condition number of the iteration operator is bounded from above by , where C is a positive constant independent of the degree N and the number of subdomains, and is the inf-sup condition of the pair -. We also consider the stationary Navier-Stokes equations; in each Newton step, a non-symmetric indefinite problem is solved using a Schwarz preconditioner. By using an especially designed low-order global space, and the solution of local problems analogous to those decribed above for the Stokes equation, we are able to present a complete theory for the method. We prove that the number of iterations of the GMRES method, at each Newton step, is bounded from above by . The constant C does not depend on the number of subdomains or N, and it does not deteriorate as the Newton iteration proceeds. Received March 2, 1998 / Revised version received October 12, 1999 / Published online March 20, 2001     

Coupling of viscous and inviscid Stokes equations via a domain decomposition method for finite elements

Summary A generalized Stokes problem is addressed in the framework of a domain decomposition method, in which the physical computational domain is partitioned into two subdomains ₁ and ₂.Three different situations are covered. In the former, the viscous terms are kept in both subdomains. Then we consider the case in which viscosity is dropped out everywhere in . Finally, a hybrid situation in which viscosity is dropped out only in ₁ is addressed. The latter is motivated by physical applications.In all cases, correct transmission conditions across the interface between ₁ and ₂ are devised, and an iterative procedure involving the successive resolution of two subproblems is proposed.The numerical discretization is based upon appropriate finite elements, and stability and convergence analysis is carried out.We also prove that the iteration-by-subdomain algorithms which are associated with the various domain decomposition approaches converge with a rate independent of the finite element mesh size.This work was partially supported by CIRA S.p.A. under the contract Coupling of Euler and Navier-Stokes equations in hypersonic flowsDeceased     

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     

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.     

A multigrid method for constrained optimal control problems

We consider the fast and efficient numerical solution of linear-quadratic optimal control problems with additional constraints on the control. Discretization of the first-order conditions leads to an indefinite linear system of saddle point type with additional complementarity conditions due to the control constraints. The complementarity conditions are treated by a primal-dual active set strategy that serves as outer iteration. At each iteration step, a KKT system has to be solved. Here, we develop a multigrid method for its fast solution. To this end, we use a smoother which is based on an inexact constraint preconditioner.We present numerical results which show that the proposed multigrid method possesses convergence rates of the same order as for the underlying (elliptic) PDE problem. Furthermore, when combined with a nested iteration, the solver is of optimal complexity and achieves the solution of the optimization problem at only a small multiple of the cost for the PDE solution.     

Two-grid discretization schemes for nonlinear Schrödinger equations

We study efficient two-grid discretization schemes with two-loop continuation algorithms for computing wave functions of two-coupled nonlinear Schrödinger equations defined on the unit square and the unit disk. Both linear and quadratic approximations of the operator equations are exploited to derive the schemes. The centered difference approximations, the six-node triangular elements and the Adini elements are used to discretize the PDEs defined on the unit square. The proposed schemes also can compute stationary solutions of parameter-dependent reaction–diffusion systems. Our numerical results show that it is unnecessary to perform quadratic approximations.     

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.     

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     

A multigrid method for Reissner-Mindlin plates

Summary The numerical solution of the Mindlin-Reissner plate equations by a multigrid method is studied. Difficulties arise only if the thickness parameter is significantly smaller than the mesh parameter. In this case an augmented Lagrangian method is applied to transform the given problem into a sequence of problems with relaxed penalty parameter. With this a parameter independent iteration is obtained.     

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.     

An adaptive multigrid scheme for Bose-Einstein condensates in a periodic potential

We present a novel multigrid-continuation method for treating parameter-dependent problems. The proposed algorithm which can be flexibly implemented is a generalization of the two-grid discretization schemes [C.-S. Chien, B.-W. Jeng, A two-grid discretization scheme for semilinear elliptic eigenvalue problems, SIAM J. Sci. Comput. 27 (2006) 1287-1304]. That is, approximating points on a solution curve do not necessarily lie on the same fine grid. We apply the algorithm to compute energy levels and superfluid densities of Bose-Einstein condensates (BEC) in a periodic potential. Both positive and negative scattering lengths are considered in our numerical experiments. For positive scattering length, if the chemical potential is large enough, and the domain is properly chosen, the results show that the number of peaks of the first few energy states of the 2D BEC in a periodic potential depends on the wave number of the periodic potential. Moreover, for bright solitons the number of peaks of the ground state solutions is and , where the periodic potential is expressed in terms of the sine or the cosine functions, respectively. However, these formulae do not hold if the scattering length is negative. The numerical study is extended to the two-component, 1D and 2D BEC in a periodic potential.     

A cascadic multigrid algorithm for semilinear elliptic problems

Summary. We propose a cascadic multigrid algorithm for a semilinear elliptic problem. The nonlinear equations arising from linear finite element discretizations are solved by Newton's method. Given an approximate solution on the coarsest grid on each finer grid we perform exactly one Newton step taking the approximate solution from the previous grid as initial guess. The Newton systems are solved iteratively by an appropriate smoothing method. We prove that the algorithm yields an approximate solution within the discretization error on the finest grid provided that the start approximation is sufficiently accurate and that the initial grid size is sufficiently small. Moreover, we show that the method has multigrid complexity. Received February 12, 1998 / Revised version received July 22, 1999 / Published online June 8, 2000     

Robust multigrid method for the planar linear elasticity problems

We consider the solution of the system of equations that arise from the higher order conforming finite element (Scott–Vogelius element) discretizations of the boundary value problems associated with the differential operator −ρ ² Δ − κ ²∇div, where ρ and κ are nonzero parameters. Robust multigrid method is constructed, i.e., the convergence rate of multigrid method is optimal with respect to the mesh size, the number of levels, and weights on the two terms in the aforementioned differential operator.

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Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: space discretization and convergence

This paper focuses on the numerical analysis of a finite element method with stabilization for the unsteady incompressible Navier–Stokes equations. Incompressibility and convective effects are both stabilized adding an interior penalty term giving L ²-control of the jump of the gradient of the approximate solution over the internal faces. Using continuous equal-order finite elements for both velocities and pressures, in a space semi-discretized formulation, we prove convergence of the approximate solution. The error estimates hold irrespective of the Reynolds number, and hence also for the incompressible Euler equations, provided the exact solution is smooth.     

A posteriori error estimators for the Stokes equations

Summary We present two a posteriori error estimators for the mini-element discretization of the Stokes equations. One is based on a suitable evaluation of the residual of the finite element solution. The other one is based on the solution of suitable local Stokes problems involving the residual of the finite element solution. Both estimators are globally upper and locally lower bounds for the error of the finite element discretization. Numerical examples show their efficiency both in estimating the error and in controlling an automatic, self-adaptive mesh-refinement process. The methods presented here can easily be generalized to the Navier-Stokes equations and to other discretization schemes.This work was accomplished at the Universität Heidelberg with the support of the Deutsche Forschungsgemeinschaft     

A posteriori error estimators for the Stokes equations II non-conforming discretizations

Summary We present an a posteriori error estimator for the non-conforming Crouzeix-Raviart discretization of the Stokes equations which is based on the local evaluation of residuals with respect to the strong form of the differential equation. The error estimator yields global upper and local lower bounds for the error of the finite element solution. It can easily be generalized to the stationary, incompressible Navier-Stokes equations and to other non-conforming finite element methods. Numerical examples show the efficiency of the proposed error estimator.