A New Extrapolation Economy Cascadic Multigrid Method for Image Restoration Problems

In this paper, a new extrapolation economy cascadic multigrid method is proposed to solve the image restoration model. The new method combines the new extrapolation formula and quadratic interpolation to design a nonlinear prolongation operator, which provides more accurate initial values for the fine grid level. An edge preserving denoising operator is constructed to remove noise and preserve image edges. The local smoothing operator reduces the influence of staircase effect. The experiment results show that the new method not only improves the computational efficiency but also ensures good recovery quality.

A Preconditioned Multigrid Method for Efficient Simulation of Three-dimensional Compressible and Incompressible Flows

To develop an efficient and robust aerodynamic analysis method for numerical optimization designs of wing and complex configuration, a combination of matrix preconditioning and multigrid method is presented and investigated. The time derivatives of three-dimensional Navier-Stokes equations are preconditioned by Choi-Merkle preconditioning matrix that is originally designed for two-dimensional low Mach number viscous flows. An extension to three-dimensional viscous flow is implemented, and a method improving the convergence for transonic flow is proposed. The space discretizaition is performed by employing a finite-volume cell-centered scheme and using a central difference. The time marching is based on an explicit Rtmge-Kutta scheme proposed by Jameson. An efficient FAS multigrid method is used to accelerate the convergence to steady-state solutions. Viscous flows over ONERA M6 wing and M100 wing are numerically simulated with Mach numbers ranging from 0.010 to 0.839. The inviscid flow over the DLR-F4 wing-body configuration is also calculated to preliminarily examine the performance of the presented method for complex configuration. The computed results are compared with the experimental data and good agreement is achieved. It is shown that the presented method is efficient and robust for both compressible and incompressible flows and is very attractive for aerodynamic optimization designs of wing and complex configuration.

THE MULTIGRID METHOD FOR RESERVOIR SIMULATION

This paper describes a way of solving the reservoir simulation pressure equation using mulligrid technique. The subroutine MG of four-grid method is presented. The result for 2-D two-phase problem is exactly the same as that of the SOR method and the CPU time is much less than that of the latter one.

Geometric Multigrid Method for Isogeometric Analysis

The isogeometric analysis method(IGA)is a new type of numerical method solving partial differential equations.Compared with the traditional finite element method,IGA based on geometric spline can keep the model consistency between geometry and analysis,and provide higher precision with less freedom.However,huge stiffness matrix fromthe subdivision progress still leads to the solution efficiency problems.This paper presents amultigrid method based on geometric multigrid(GMG)to solve the matrix system of IGA.This method extracts the required computational data for multigrid method fromthe IGA process,which also can be used to improve the traditional algebraic multigrid method(AGM).Based on this,a full multigrid method(FMG)based on GMG is proposed.In order to verify the validity and reliability of these methods,this paper did some test on Poisson’s equation and Reynolds’equation and compared the methods on different subdivision methods,different grid degrees of freedom,different cyclic structure degrees,and studied the convergence rate under different subdivision strategies.The results show that the proposed method is superior to the conventional algebraic multigrid method,and for the standard relaxed V-cycle iteration,the method still has a convergence speed independent of the grid size at the same degrees.

The V-cycle Multigrid Method for a Hermite-type Rectangular Element

In this paper, a V-cycle multigrid method is presented for a Hermite rectangular element. By defining proper mesh-dependent inner product and transfer operator, we obtain its convergence property and the uniform convergence rate independent of mesh size and level are established.

Multigrid Method for the Numerical Solution of the Modified Equal Width Wave Equation

Numerical solutions of the modified equal width wave equation are obtained by using the multigrid method and finite difference method. The motion of a single solitary wave, interaction of two solitary waves and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. The numerical solutions are compared with the known analytical solutions. Using error norms and conservative properties of mass, momentum and energy, accuracy and efficiency of the mentioned method will be established through comparison with other methods.

AN OPTIMAL V-CYCLE MULTIGRID METHOD FOR CONFORMING AND NONCONFORMING PLATE ELEMENTS

In this paper, an optimal V-cycle multigrid method for some conforming and nonconforming plate elements are constructed. A new method dealing with nonnested multigrid methods is presented.

CASCADIC MULTIGRID METHODS FOR MORTAR WILSON FINITE ELEMENT METHODS ON PLANAR LINEAR ELASTICITY

Cascadic multigrid technique for mortar Wilson finite element method ofhomogeneous boundary value planar linear elasticity is described and analyzed. Firstthe mortar Wilson finite element method for planar linear elasticity will be analyzed,and the error estimate under L2 and H1 norm is optimal. Then a cascadic multigridmethod for the mortar finite element discrete problem is described. Suitable grid trans-fer operator and smoother are developed which lead to an optimal cascadic multigridmethod. Finally, the computational results are presented.

The Numerical Solution of the MRLW Equation Using the Multigrid Method

In this paper, we obtained the numerical solutions of the modified regularized long-wave (MRLW) equation, by using the multigrid method and finite difference method. The solitary wave motion, interaction of two and three solitary waves, and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. The numerical solutions are compared with the known analytical solutions. Usingerror norms and conservative properties of mass, momentum and energy, accuracy and efficiency of the mentioned method will be established through comparison with other techniques.

High Order Compact Difference Scheme and Multigrid Method for 2D Elliptic Problems with Variable Coefficients and Interior/Boundary Layers on Nonuniform Grids

In this paper, a high order compact difference scheme and a multigrid method are proposed for solving two-dimensional (2D) elliptic problems with variable coefficients and interior/boundary layers on nonuniform grids. Firstly, the original equation is transformed from the physical domain (with a nonuniform mesh) to the computational domain (with a uniform mesh) by using a coordinate transformation. Then, a fourth order compact difference scheme is proposed to solve the transformed elliptic equation on uniform girds. After that, a multigrid method is employed to solve the linear algebraic system arising from the difference equation. At last, the numerical experiments on some elliptic problems with interior/boundary layers are conducted to show high accuracy and high efficiency of the present method.

A Newton multigrid method for steady-state shallow water equations with topography and dry areas

A Newton multigrid method is developed for one-dimensional （1D） and two- dimensional （2D） steady-state shallow water equations （SWEs） with topography and dry areas. The nonlinear system arising from the well-balanced finite volume discretization of the steady-state SWEs is solved by the Newton method as the outer iteration and a geometric multigrid method with the block symmetric Gauss-Seidel smoother as the inner iteration. The proposed Newton multigrid method makes use of the local residual to regularize the Jacobian matrix of the Newton iteration, and can handle the steady- state problem with wet/dry transition. Several numerical experiments are conducted to demonstrate the efficiency, robustness, and well-balanced property of the proposed method. The relation between the convergence behavior of the Newton multigrid method and the distribution of the eigenvalues of the iteration matrix is detailedly discussed.

Cascadic multigrid methods for parabolic problems

In this paper,we consider the cascadic multigrid method for a parabolic type equation.Backward Euler approximation in time and linear finite element approximation in space are employed.A stability result is established under some conditions on the smoother.Using new and sharper estimates for the smoothers that reflect the precise dependence on the time step and the spatial mesh parameter,these conditions are verified for a number of popular smoothers.Optimal error bound sare derived for both smooth and non-smooth data.Iteration strategies guaranteeing both the optimal accuracy and the optimal complexity are presented.

A full multigrid method for nonlinear eigenvalue problems

We introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of the nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and nonlinear eigenvalue problems on the coarsest finite element space. The linearized boundary value problems are solved by some multigrid iterations.Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as the linear problem solver. We prove that the computational work of this new scheme is truly optimal,the same as solving the linear corresponding boundary value problem. In this case, this type of iteration scheme certainly improves the overfull efficiency of solving nonlinear eigenvalue problems. Some numerical experiments are presented to validate the efficiency of the new method.

GAUSS-SEIDEL-TYPE MULTIGRID METHODS

By making use of the Gauss-Seidel-type solution method, the procedure for computing the interpolation operator of multigrid methods is simplified. This leads to a saving of computational time. Three new kinds of interpolation formulae are obtained by adopting different approximate methods, to try to enhance the accuracy of the interpolatory operator. A theoretical study proves the two-level convergence of these Gauss-Seidel-type MG methods. A series of numerical experiments is presented to evaluate the relative performance of the methods with respect to the convergence factor, CPU-time(for one V-cycle and the setup phase) and computational complexity.

Extrapolation cascadic multigrid method on piecewise uniform grid

The triangular linear finite elements on piecewise uniform grid for an elliptic problem in convex polygonal domain are discussed. Global superconvergence in discrete Hi-norm and global extrapolation in discrete L2-norm are proved. Based on these global estimates the conjugate gradient method （CG） is effective, which is applied to extrapolation cascadic multigrid method （EXCMG）. The numerical experiments show that EXCMG is of the global higher accuracy for both function and gradient.

CASCADIC MULTIGRID METHOD FOR THE MORTAR ELEMENT METHOD FOR P1 NONCONFORMING ELEMENT

In this paper, we consider the cascadic multigrid method for the mortar P1 nonconforming element which is used to solve the Poisson equation and prove that the cascadic conjugate gradient method is accurate with optimal complexity.

PRECONDITIONING HIGHER ORDER FINITE ELEMENT SYSTEMS BY ALGEBRAIC MULTIGRID METHOD OF LINEAR ELEMENTS

We present and analyze a robust preconditioned conjugate gradient method for the higher order Lagrangian finite element systems of a class of elliptic problems. An auxiliary linear element stiffness matrix is chosen to be the preconditioner for higher order finite elements. Then an algebraic multigrid method of linear finite element is applied for solving the preconditioner. The optimal condition number which is independent of the mesh size is obtained. Numerical experiments confirm the efficiency of the algorithm.

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.

Multigrid Method for Poroelasticity Problem by Finite Element Method

In this paper,we will investigate a multigrid algorithm for poroelasticity problem by a new finite element method with homogeneous boundary conditions in two dimensional space.We choose N´ed´elec edge element for the displacement variable and piecewise continuous polynomials for the pressure variable in the model problem.In constructing multigrid algorithm,a distributive Gauss-Seidel iteration method is applied.Numerical experiments shows that the finite element method achieves optimal convergence order and the multigrid algorithm is almost uniformly convergent to mesh size h and parameter dt on regular meshes.

A Shifted-Inverse Adaptive Multigrid Method for the Elastic Eigenvalue Problem

A shifted-inverse iteration is proposed for the finite element discretization of the elastic eigenvalue problem.The method integrates the multigrid scheme and adaptive algorithm to achieve high efficiency and accuracy.Error estimates and optimal convergence for the proposed method are proved.Numerical examples show that the proposed method inherits the advantages of both ingredients and can compute low regularity eigenfunctions effectively.