Jacobi Relaxation Research Articles

A Vanka‐based parameter‐robust multigrid relaxation for the Stokes–Darcy Brinkman problems

AbstractWe consider a block‐structured multigrid method based on Braess–Sarazin relaxation for solving the Stokes–Darcy Brinkman equations discretized by the marker and cell scheme. In the relaxation scheme, an element‐based additive Vanka operator is used to approximate the inverse of the corresponding shifted Laplacian operator involved in the discrete Stokes–Darcy Brinkman system. Using local Fourier analysis, we present the stencil for the additive Vanka smoother and derive an optimal smoothing factor for Vanka‐based Braess–Sarazin relaxation for the Stokes–Darcy Brinkman equations. Although the optimal damping parameter is dependent on meshsize and physical parameter, it is very close to one. In practice, we find that using three sweeps of Jacobi relaxation on the Schur complement system is sufficient. Numerical results of two‐grid and V(1,1)‐cycle are presented, which show high efficiency of the proposed relaxation scheme and its robustness to physical parameters and the meshsize. Using a damping parameter equal to one gives almost the same convergence results as these for the optimal damping parameter.

Smoothing Analysis of Two Robust Multigrid Methods for Elliptic Optimal Control Problems

In this paper we study and compare two multigrid relaxation schemes with coarsening by two, three, and four for solving elliptic sparse optimal control problems with control constraints and combined and cost functional. First, we perform a detailed local Fourier analysis (LFA) of a well-known collective Jacobi relaxation (CJR) scheme for the unconstrained case with only cost functional, where the optimal smoothing factors are derived. This insightful analysis reveals that the optimal relaxation parameters depend on both the mesh step size and the regularization parameter , which was not investigated in literature. Second, we propose and analyze a new mass-based Braess--Sarazin relaxation (BSR) scheme, which is proven to provide smaller smoothing factors than the CJR scheme when for a small constant . Finally, these schemes are successfully extended to control-constrained cases through the semismooth Newton method, where the corresponding Jacobian systems are treated by the proposed multigrid schemes. The nonstandard coarsening by three or four with BSR is competitive with the standard coarsening by two. Numerical examples are presented to validate our theoretical outcomes. The proposed inexact BSR (IBSR) scheme, where two preconditioned conjugate gradients (PCG) iterations are applied to solve the Schur complement system, yields a better computational efficiency than the CJR scheme in the conducted numerical comparison.

On a Multigrid Method for Tempered Fractional Diffusion Equations

In this paper, we develop a suitable multigrid iterative solution method for the numerical solution of second- and third-order discrete schemes for the tempered fractional diffusion equation. Our discretizations will be based on tempered weighted and shifted Grünwald difference (tempered-WSGD) operators in space and the Crank–Nicolson scheme in time. We will prove, and show numerically, that a classical multigrid method, based on direct coarse grid discretization and weighted Jacobi relaxation, performs highly satisfactory for this type of equation. We also employ the multigrid method to solve the second- and third-order discrete schemes for the tempered fractional Black–Scholes equation. Some numerical experiments are carried out to confirm accuracy and effectiveness of the proposed method.

Multigrid preconditioners for anisotropic space-fractional diffusion equations

We focus on a two-dimensional time-space diffusion equation with fractional derivatives in space. The use of Crank-Nicolson in time and finite differences in space leads to dense Toeplitz-like linear systems. Multigrid strategies that exploit such structure are particularly effective when the fractional orders are both close to 2. We seek to investigate how structure-based multigrid approaches can be efficiently extended to the case where only one of the two fractional orders is close to 2, i.e., when the fractional equation shows an intrinsic anisotropy. Precisely, we design a multigrid (block-banded–banded-block) preconditioner whose grid transfer operator is obtained with a semi-coarsening technique and that has relaxed Jacobi as smoother. The Jacobi relaxation parameter is estimated by using an automatic symbol-based procedure. A further improvement in the robustness of the proposed multigrid method is attained using the V-cycle with semi-coarsening as smoother inside an outer full-coarsening. Several numerical results confirm that the resulting multigrid preconditioner is computationally effective and outperforms current state of the art techniques.

Hyperbolic relaxation method for elliptic equations

We show how the basic idea of parabolic Jacobi relaxation can be modified to obtain a new class of hyperbolic relaxation schemes that are suitable for the solution of elliptic equations. Some of the analytic and numerical properties of hyperbolic relaxation are examined. We describe its implementation as a first order system in a pseudospectral evolution code, demonstrating that certain elliptic equations can be solved within a framework for hyperbolic evolution systems. Applications include various initial data problems in numerical general relativity. In particular we generate initial data for the evolution of a massless scalar field, a single neutron star, and binary neutron star systems.

Block-accelerated aggregation multigrid for Markov chains with application to PageRank problems

Recently, the adaptive algebraic aggregation multigrid method has been proposed for computing stationary distributions of Markov chains. This method updates aggregates on every iterative cycle to keep high accuracies of coarse-level corrections. Accordingly, its fast convergence rate is well guaranteed, but often a large proportion of time is cost by aggregation processes. In this paper, we show that the aggregates on each level in this method can be utilized to transfer the probability equation of that level into a block linear system. Then we propose a Block–Jacobi relaxation that deals with the block system on each level to smooth error. Some theoretical analysis of this technique is presented, meanwhile it is also adapted to solve PageRank problems. The purpose of this technique is to accelerate the adaptive aggregation multigrid method and its variants for solving Markov chains and PageRank problems. It also attempts to shed some light on new solutions for making aggregation processes more cost-effective for aggregation multigrid methods. Numerical experiments are presented to illustrate the effectiveness of this technique.

Smoothing Analysis of Distributive Red-Black Jacobi Relaxation for Solving 2D Stokes Flow by Multigrid Method

Smoothing analysis process of distributive red-black Jacobi relaxation in multigrid method for solving 2D Stokes flow is mainly investigated on the nonstaggered grid by using local Fourier analysis (LFA). For multigrid relaxation, the nonstaggered discretizing scheme of Stokes flow is generally stabilized by adding an artificial pressure term. Therefore, an important problem is how to determine the zone of parameter in adding artificial pressure term in order to make stabilization of the algorithm for multigrid relaxation. To end that, a distributive red-black Jacobi relaxation technique for the 2D Stokes flow is established. According to the 2h-harmonics invariant subspaces in LFA, the Fourier representation of the distributive red-black Jacobi relaxation for discretizing Stokes flow is given by the form of square matrix, whose eigenvalues are meanwhile analytically computed. Based on optimal one-stage relaxation, a mathematical relation of the parameter in artificial pressure term between the optimal relaxation parameter and related smoothing factor is well yielded. The analysis results show that the numerical schemes for solving 2D Stokes flow by multigrid method on the distributive red-black Jacobi relaxation have a specified convergence parameter zone of the added artificial pressure term.

A Smoothing Process of Multicolor Relaxation for Solving Partial Differential Equation by Multigrid Method

This paper is concerned with a novel methodology of smoothing analysis process of multicolor point relaxation by multigrid method for solving elliptically partial differential equations (PDEs). The objective was firstly focused on the two-color relaxation technique on the local Fourier analysis (LFA) and then generalized to the multicolor problem. As a key starting point of the problems under consideration, the mathematical constitutions among Fourier modes with various frequencies were constructed as a base to expand two-color to multicolor smoothing analyses. Two different invariant subspaces based on the 2h-harmonics for the two-color relaxation with two and four Fourier modes were constructed and successfully used in smoothing analysis process of Poisson’s equation for the two-color point Jacobi relaxation. Finally, the two-color smoothing analysis was generalized to the multicolor smoothing analysis problems by multigrid method based on the invariant subspaces constructed.

P = 2 Continuous finite elements on tetrahedra with local mass matrix inversion

When using continuous finite element methods for unsteady simulations, one must invert a global mass matrix even for an explicit time advancement scheme. In this paper we develop a local mass matrix inversion procedure for p=2, C0 tetrahedral elements and also show that, unlike on triangles [B.T. Helenbrook, SIAM J. Numer. Anal., 47 (2009) 1304–1318], this procedure does not generalize to higher order. For the p=2 basis, it is demonstrated that the accuracy of the local inversion is better than any previous mass lumping scheme. This allows the development of fast and low memory iterative algorithms as well as accurate explicit computations to be performed. The efficiency of using the approach as an iterative relaxation scheme for solving the Helmholtz equation is shown by comparing to more standard approaches such as p-multigrid with Jacobi relaxation. Lastly, the local mass matrix inversion is used to march the unsteady heat equation explicitly with third order spatial accuracy.

Algebraic multigrid and 4th‐order discrete‐difference equations of incompressible fluid flow

AbstractThis paper investigates the effectiveness of two different Algebraic Multigrid (AMG) approaches to the solution of 4th‐order discrete‐difference equations for incompressible fluid flow (in this case for a discrete, scalar, stream‐function field). One is based on a classical, algebraic multigrid, method (C‐AMG) the other is based on a smoothed‐aggregation method for 4th‐order problems (SA‐AMG). In the C‐AMG case, the inter‐grid transfer operators are enhanced using Jacobi relaxation. In the SA‐AMG case, they are improved using a constrained energy optimization of the coarse‐grid basis functions. Both approaches are shown to be effective for discretizations based on uniform, structured and unstructured, meshes. They both give good convergence factors that are largely independent of the mesh size/bandwidth. The SA‐AMG approach, however, is more costly both in storage and operations. The Jacobi‐relaxed C‐AMG approach is faster, by a factor of between 2 and 4 for two‐dimensional problems, even though its reduction factors are inferior to those of SA‐AMG.For non‐uniform meshes, the accuracy of this particular discretization degrades from 2nd to 1st order and the convergence factors for both methods then become mesh dependent. Copyright © 2009 John Wiley & Sons, Ltd.

Application of p-Multigrid to Discontinuous Galerkin Formulations of the Euler Equations

We investigate p-multigrid as a solution method for several different discontinuous Galerkin (DG) formulations of the Poisson equation. Different combinations of relaxation schemes and basis sets have been combined with the DG formulations to find the best performing combination. The damping factors of the schemes have been determined using Fourier analysis for both one and two-dimensional problems. One important finding is that when using DG formulations, the standard approach of forming the coarse p matrices separately for each level of multigrid is often unstable. To ensure stability the coarse p matrices must be constructed from the fine grid matrices using algebraic multigrid techniques. Of the relaxation schemes, we find that the combination of Jacobi relaxation with the spectral element basis is fairly effective. The results using this combination are p sensitive in both one and two dimensions, but reasonable convergence rates can still be achieved for moderate values of p and isotropic meshes. A competitive alternative is a block Gauss-Seidel relaxation. This actually out performs a more expensive line relaxation when the mesh is isotropic. When the mesh becomes highly anisotropic, the implicit line method and the Gauss-Seidel implicit line method are the only effective schemes. Adding the Gauss-Seidel terms to the implicit line method gives a significant improvement over the line relaxation method.

Fourier Analysis of Multigrid Methods on Hexagonal Grids

This paper applies local Fourier analysis to multigrid methods on hexagonal grids. Using oblique coordinates to express the grids and a dual basis for the Fourier modes, the analysis proceeds essentially the same as for rectangular grids. The framework for one- and two-grid analyses is given and then applied to analyze the performance of multigrid methods for the Poisson problem on a hexagonal grid. Numerical results confirm the analysis. Uniform hexagonal grids provide an approximation to spherical geodesic grids; numerical results for the latter show similar performance. While the analysis is similar to that for rectangular grids, the results differ somewhat: full weighting is superior to injection for restriction, Jacobi relaxation performs about as well as Gauss–Seidel relaxation, and underrelaxation is not required for good performance. Also, coarse-fine or four-color ordering (both analogues of red-black ordering on the rectangular grid) improves the performance of Jacobi relaxation, with the latter achieving a smoothing factor of approximately 0.25. An especially simple compact fourth-order discretization works well, and the full multigrid algorithm produces the solution to the level of truncation error in work proportional to the number of unknowns.

An efficient semi-coarsening multigrid method for variable diffusion problems in cylindrical coordinates

In this paper, we present an efficient multigrid (MG) algorithm for solving the three-dimensional variable coefficient diffusion equation in cylindrical coordinates. The multigrid V-cycle combines a semi-coarsening in azimuthal direction with the red-black Gauss–Seidel plane (radial-axial plane) relaxation. On each plane relaxation, we further semi-coarsen the axial direction with red-black line relaxation in the radial direction. We also prove the convergence of two-level MG with plane Jacobi relaxation. Numerical results show that the present multigrid method indeed is scalable with the mesh size.

Spectral Preconditioners for Nonhydrostatic Atmospheric Models

Abstract The elliptic problems in semi-implicit nonhydrostatic atmospheric models are difficult. Typically, they are poorly conditioned, nonseparable, contain cross-derivative terms, and are often nonsymmetric. Here, the resulting linear system is solved using a preconditioned Krylov subspace method—the generalized conjugate residual (GCR) algorithm. A horizontal spectral preconditioner is developed as an alternative to a more standard and much simpler line Jacobi relaxation scheme. However, the efficacy of the spectral preconditioner requires neglecting the cross-derivative terms and homogenization (e.g., averaging) metric coefficients over the computational domain. Because such a compromise causes a substantial departure of the preconditioner from the original elliptic operator, it is not obvious a priori whether it leads to a competitive solver. The robustness of the proposed approach over a broad range of representative meteorological applications is evaluated, in the context of a three-time-level sem...

Improved Convergence for CCD Gain Calibration Using Simultaneous-Overrelaxation Techniques

Simple modifications to the analysis used in the Kuhn, Lin, & Loranz flat-field CCD calibration method yield significant improvements in both speed and accuracy. In this method, multiple exposures are taken of a time-independent signal at different spatial positions. The flat field is then expressed in the form of a Jacobi relaxation solution to Poisson's equation. By applying the technique of simultaneous overrelaxation, we have improved the convergence rate to require approximately the square root of the number of iterations () needed by the Jacobi method. For large arrays, where r is correspondingly large, this improvement is significant. Furthermore, we have improved the accuracy by extending the method to account for fractional pixel shifts.

Multigrid computation for micromagnetics

A numerical scheme is presented in which a static micromagnetic calculation is performed using one of the multigrid methods, the V-cycle method. An equilibrium equation in which the discretized Laplacian of magnetization is solved is derived from the magnetic energy and relaxed in the V-cycle method based on the weighted Jacobi relaxation. Calculations made for a one-dimension Bloch wall and a flux-closure structure in Permalloy™ films show the validity and the versatility of the new scheme.

Multigrid smoothing for symmetric nine-point stencils

Analytical formulae are obtained for the smoothing factors yielded by damped Jacobi relaxation and by red-black relaxation applied to symmetric nine-point stencil discretizations of elliptic partial differential operators in 2D. The results include point and line relaxation and full and partial coarsening. Several unusual results are implied by the formulae. Numerical results of multigrid cycles match the analytical predictions well.

Nonlinear static stress analysis of submarine high pressure pipelines

The structural behaviour of submarine natural gas transmission pipelines is simulated with the aid of a finite difference numerical model. A conditional point Jacobi relaxation method is adopted for the simultaneous identification of the pipe deformation and the unknown contact points along the initially assumed rigid seabed profile. A method using the linear spring concept is proposed for examining the subsequent vertical soil displacements. The pipe gradients and the tensile forces developed due to pressurization are the main sources of non-linearity considered in the fundamental equation of flexure. Maximum probable hydro-dynamic lift forces are taken into account during the empty pipeline phase analysis. A wide range of loading schemes is investigated in order to ascertain that the bending stresses remain within the acceptable range for the pipe material. Due to a high probability of soil liquefaction occurrence during earthquakes, a stress analysis is carried out using a suitably altered seabed profile in order to identify the points of critical bending.

A Non-Linear Aeroelasticity Analysis of a Fan Blade Using Unstructured Dynamic Meshes

The main objective of this paper is to present a methodology for the three-dimensional aeroelasticity analysis of turbomachinery blades using an unstructured compressible Navier-Stokes solver for the fluid and a modal model for the structure. The basic fluid solver is constructed in the form of a central difference scheme with explicitly added artificial dissipation which is based upon the fourth- and second-order differences of the solution. The temporal discretization uses an implicit time integration scheme based on a Jacobi relaxation procedure. The structural modes of vibration are determined via a finite element model and the mode shapes are interpolated on to the fluid mesh in a manner that is consistent with general unstructured tetrahedral grids. A spring analogy algorithm that can move the mesh according to the instantaneous shape of a deforming blade has been developed for the accurate tracking of the solid boundaries without creating excessive grid distortions. The performance of the resulting system was examined by considering the aeroelastic behaviour of NASA Rotor 67 fan blade and predictions were compared to experimental results wherever possible. Using a three-dimensional cyclic symmetry model, the tip leading edge time histories were predicted under peak-efficiency and near-stall conditions, and the corresponding aeroelastic natural frequencies and aerodynamic damping values were determined. The blade was found to be stable in all cases considered.

A multigrid method with unstructured adaptive grids for steady Euler equations

The multigrid method based on multi-stage Jacobi relaxation, earlier developed by the authors for structured grid calculations with Euler equations, is extended to unstructured grid applications. The meshes are generated with Delaunay triangulation algorithms and are adapted to the flow solution.