A Fourier Transform Analysis of Convergence Properties of Multigrid V-Cycle Algorithm

Free vibration analysis of cylindrical shells using the Haar wavelet method

After discretization of the partial differential equations from mechanics, one usually obtains large systems of (non)linear equations. Their efficient solution requires the use of fast iterative methods. Multigrid iterations are able to solve linear and nonlinear systems with a rather fast rate of convergence, provided the problem is of elliptic type. The contribution describes the basic construction of multigrid methods, their ingredients, related methods, and their application to various problem classes.Although most of the applications concern FEM discretizations of partial differential equations, there are also applications to integral equations as they occur in boundary element methods (BEM).

For the numerical solution of linear systems that arise from discretized linear partial differential equations, multigrid and domain decomposition methods are well established. Multigrid methods are known to have optimal complexity and domain decomposition methods are in particular useful for parallelization of the implemented algorithm. For linear random operator equations, the classical theory is not directly applicable, since condition numbers of system matrices may be close to degenerate due to non-uniform random input. It is shown that iterative methods converge in the strong, i.e. $$L^p$$, sense if the random input satisfies certain integrability conditions. As a main result, standard multigrid and domain decomposition methods are applicable in the case of linear elliptic partial differential equations with lognormal diffusion coefficients and converge strongly with deterministic bounds on the computational work which are essentially optimal. This enables the application of multilevel Monte Carlo methods with rigorous, deterministic bounds on the computational work.

Multigrid (MG) methods are presented for fast, efficient, flexible, and robust least-squares wavefront reconstruction in extremely high-resolution conventional adaptive optics, or ExAO. We demonstrate that MG can robustly handle a variety of sensor-actuator geometries, and it can accommodate deformable mirror influence function models that are more realistic than the common piecewise bilinear model. With MG one can also easily incorporate additional penalty, or regularization, terms to damp out the waffle mode in Fried geometry and to damp out instabilities due to actuators near the pupil boundary with poorly sensed influence. We present closed-loop simulation results that suggest that only one or two MG iterations per time step are needed to control an ExAO system.

In this paper we discuss the application of a finite-volume multigrid method to solve three-dimensional thermally driven convection in a highly viscous, incompressible fluid with a variable viscosity. The conservation laws are solved in the primitive variable formulation, A second-order control volume method is used as discretization. Two schemes are used for time stepping, a second-order implicit-explicit scheme based on the Crank-Nicolson and Adams-Bashforth method, and a fully-implicit θ-method. The implicit system of nonlinear equations are solved using multigrid iteration with the SIMPLER method as smoother. In this paper, we describe the implemented multigrid method and investigate its efficiency and the robustness for different viscosity contrasts. Convergence tests showed that with a small modification of the SIMPLER method, the multigrid method exhibits a satisfactory convergence rate even for viscosity contrasts up to 109. Three cases of time-dependent thermally driven convection with v...

The paper describes the development and application of a new multigrid method using adaptive-prismatic grids for viscous flow computations. The three-dimensional Navier-Stokes equations are solved on the prismatic grids that are adaptively refined locally. The multigrid method is employed to propagate the fine grid corrections more rapidly by redistributin g the changes in time of the solution from the fine grid to the coarser grids to accelerate convergence. The present approach uses the parent cells of the fine grid cells in an adapted mesh to generate successively coarser levels of multigrid. Furthermore, the prismatic grid is semi-unstruct ured and is constituted by layers of cells that are constructed from the triangulation on the surface of the body in a direction normal to the surface. This inherent structure of the prismatic grid is used in generating further coarser multigrid levels by deleting every other layer of cells. The solver is an explicit, vertex-based, finite volume scheme. Spatial discretization is of central-differencing type and temporal discretization is of Lax-Wendroff type. Application cases include adaptive solutions obtained with multigrid acceleration for flows with both shear layers and shock waves present. Convergence acceleration obtained by employing various levels of multigrid is investigated.

The multigrid method widely used in computational fluid dynamics is applied to the problem of propagation of an optical wave in an optical waveguide. This method is a technique for accelerating the convergence speed of the relaxation method. The electromagnetic field components contain various frequency components. Higher‐frequency components contain local effects of the shape. On the other hand, the lower‐frequency components contain more global aspects. By using this phenomenon, the discretized equations are made to reduce errors corresponding to the grid size. As a numerical example, linear and nonlinear directional couplers are treated and the effects of the number of levels and the number of iterations in the multigrid method are studied. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 2, 86(7): 14–22, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjb.10131

We present an extrapolation multiscale multigrid (EMMG) algorithm to solve the large linear systems arising from a sixth order compact discretization of the two dimensional Poisson equation, based on multigrid method and an extrapolation operator. With the help of Taylor expansion and interpolation theory, we develop three mid-point extrapolation formulas and combine it with the classical Richardson extrapolation strategy to design an extrapolation operator. Applying this proposed extrapolation operator for the sixth order difference solutions on the finest and finer grids, which have been computed by V-cycle multigrid method, we can construct an eighth order accurate extrapolation solution on the entire finest grid directly and efficiently. Moreover, we discuss the error of EMMG method in theoretically, and conduct some numerical experiments on square or reentrant domains, to verify that our EMMG algorithm can achieve eighth order convergence and keep less cost simultaneously.

Minimum, central and average film thicknesses have been calculated for the isothermal E.H.L. point contact case, for a variety of load, rolling speed, and material parameters. The equations governing this problem were solved using a Multigrid method. This technique offers the possibility to work with a very fine grid, obtaining detailed and accurate solutions, at the cost of moderate cpu times and storage requirements, on medium size computers. Computations for low loads, requiring a large inlet zone, have been carried out using local grid refinements. The fluid in these calculations is assumed to be compressible and its viscosity-pressure behavior is described by either the Roelands equation, or the Barus equation. The ratio between the calculated minimum film thickness and the central value varied with the parameters governing the problem, but for low loads, a value of 3/4 was obtained. The film thickness behavior at these low loads can be accurately described in terms of the minimum film thickness. For higher loads, however, a description based on a film thickness, averaged over the Hertzian contact, is more appropriate to be compared with the asymptotic solution (Ertel, Grubin).

: To illustrate Brandt's multigrid method this report considers the Helmholtz equation in cases where the frequency is close to an eigenfrequency. In a second part the eigenvalue problem is discussed. The basic idea for the treatment of the inhomogeneous problem is contained in Brandt's publications. Difficulties are encountered because the coarse grid eigenvalues differ from those of the finer grid. As a consequence, approximations to a solution in the coarse grid may differ widely from the corresponding approximation in a finer grid, if the frequency defined by the inhomogeneous problem is close to one of the eigenfrequencies. This may be detrimental to the aim of the coarse grid step; namely to remove from an approximation in a fine grid long waves which cause divergence of fine grid iterations. This is remedied by eliminating long wave contributions to the residual directly in the next finer grid. The success of this procedure still depends upon the choice of the subspace from which the long wave corrections in the finer grid are taken. Even this modified method fails if the frequency pertaining to the inhomogeneous term is too close to an eigenvalue, unless the subspace contains the pertinent eigenfunction.

The numerical simulation of modern engineering problems via finite elements requires the solution of sparse linear systems of millions or even billions of unknowns. The algebraic multigrid (AMG) methods are the most common choice as linear solvers because of their fast convergence even for large-size problems. In this communication, we propose Chronos, a massively parallel implementation of a novel AMG framework, specifically designed to address complex problems by adapting its components, from the smoother, to the coarse grid correction and prolongation to the problem at hand. This work demonstrates not only the numerical performance of the proposed library, but also its robustness and adaptability to very challenging matrices, arising from different fields of application.

Fast and robust sixth-order multigrid computation for the three-dimensional convection–diffusion equation

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.

Convergence acceleration of the rational Runge-Kutta scheme for the Euler and Navier-Stokes equations

We propose to enhance the stability of IMEX schemes applied to the numerical solution of nonlinear parabolic equations by using proper smoothing procedures on bands of frequency components of the solution. The separation of those different components is realized by considering a multigrid (MG) approach. The time scheme on the coarsest grid is implicit and energy diminishing, while time marching is done by a weighted damped IMEX scheme on the finer grids. This approach allows to concentrate most computational efforts on a small set of components (the coarsest grid) and to fine-tune the damping parameters so that there is limited consistency perturbation while enhancing stability. We provide stability results and numerical illustrations on pattern dynamics, Swift-Hohenberg, and crystal-phase fields equations, in both 2D and 3D cases. The results obtained with MG are comparable with those given by a one-grid simulation.