On the robustness of a multigrid method for anisotropic reaction-diffusion problems

In this paper, we consider a reaction-diffusion boundary value problem in a three-dimensional thin domain. The very different length scales in the geometry result in an anisotropy effect. Our study is motivated by a parabolic heat conduction problem in a thin foil leading to such anisotropic reaction-diffusion problems in each time step of an implicit time integration method [7]. The reaction-diffusion problem contains two important parameters, namely ε >0 which parameterizes the thickness of the domain and μ >0 denoting the measure for the size of the reaction term relative to that of the diffusion term. In this paper we analyze the convergence of a multigrid method with a robust (line) smoother. Both, for the W- and the V-cycle method we derive contraction number bounds smaller than one uniform with respect to the mesh size and the parameters ε and μ.      

一类四阶各向异性扩散方程在图像放大中的应用

在文中我们首先分析了进行图像放大时各向异性偏微分方程优于各向同性偏微分方程,随后我们分析了在本文中不同四阶模型的扩散方向.为了消除低阶偏微分方程在处理图像中出现的块状效应的影响,同时保证方程为各向异性扩散,我们构造了两个各向异性的四阶偏微分方程,并且分别从数据和放大图像效果两方面来说明我们给出的模型优于文中提到的其它四个模型.     

A discontinuous finite difference streamline diffusion method for time-dependent hyperbolic problems

In this article, a new finite element method, discontinuous finite difference streamline diffusion method (DFDSD), is constructed and studied for first-order linear hyperbolic problems. This method combines the benefit of the discontinuous Galerkin method and the streamline diffusion finite element method. Two fully discrete DFDSD schemes (Euler DFDSD and Crank–Nicolson (CN) DFDSD) are constructed by making use of the difference discrete method for time variables and the discontinuous streamline diffusion method for space variables. The stability and optimal L² norm error estimates are established for the constructed schemes. This method makes contributions to the discontinuous methods. Finally, a numerical example is provided to show the benefit of high efficiency and simple implementation of the schemes.     

Tailored finite point method for solving one-dimensional Burgers' equation

We propose a tailored finite point method (TFPM) for solving a quasilinear time-dependent Burgers' equation with a small coefficient of viscosity. The selected basis functions for the TFPM automatically fit the properties of the local solution in time and space simultaneously. The stability and error analysis for the TFPM are given. We also demonstrate the efficiency of the proposed scheme on relatively coarse meshes. The numerical results indicate that the TFPM achieves high accuracy and effectively captures the shock solutions.     

Ultrasound speckle reduction by a SUSAN-controlled anisotropic diffusion method

An ultrasound speckle reduction method is proposed in this paper. The filter, which enhances the power of anisotropic diffusion with the Smallest Univalue Segment Assimilating Nucleus (SUSAN) edge detector, is referred to as the SUSAN-controlled anisotropic diffusion (SUSAN_AD). The SUSAN edge detector finds image features by using local information from a pseudo-global perspective. Thanks to the noise insensitivity and structure preservation properties of SUSAN, a better control can be provided to the subsequent diffusion process. To enhance the adaptability of the SUSAN_AD, the parameters of the SUSAN edge detector are calculated based on the statistics of a fully formed speckle (FFS) region. Different FFS estimation schemes are proposed for envelope-detected speckle images and log-compressed ultrasonic images. Adaptive diffusion threshold estimation and automatic diffusion termination criterion are employed to enhance the robustness of the method. Both synthetic and real ultrasound images are used to evaluate the proposed method. The performance of the SUSAN_AD is compared with four other existing speckle reduction methods. It is shown that the proposed method is superior to other methods in both noise reduction and detail preservation.     

Convergence of an upwind control-volume mixed finite element method for convection–diffusion problems

Summary We consider a upwind control volume mixed finite element method for convection–diffusion problem on rectangular grids. These methods use the lowest order Raviart–Thomas mixed finite element space as the trial functional space and associate control-volumes, or covolumes, with the vector variable as well as the scalar variable. Chou et al. [6] established a one-half order convergence in discrete L ²-norms. In this paper, we establish a first order convergence for both the vector variable as well as the scalar variable in discrete L ²-norms.      

Adaptive anisotropic meshing for steady convection-dominated problems

Obtaining accurate solutions for convection–diffusion equations is challenging due to the presence of layers when convection dominates the diffusion. To solve this problem, we design an adaptive meshing algorithm which optimizes the alignment of anisotropic meshes with the numerical solution. Three main ingredients are used. First, the streamline upwind Petrov–Galerkin method is used to produce a stabilized solution. Second, an adapted metric tensor is computed from the approximate solution. Third, optimized anisotropic meshes are generated from the computed metric tensor by an anisotropic centroidal Voronoi tessellation algorithm. Our algorithm is tested on a variety of two-dimensional examples and the results shows that the algorithm is robust in detecting layers and efficient in avoiding non-physical oscillations in the numerical approximation.     

A new iterative method for large sparse saddle point problems

In this short note, the convergence of a new iterative method for the Saddle Point Problem is presented.     

A class of nonlinear proximal point algorithms for variational inequality problems

This work is motivated by a recent work on an extended linear proximal point algorithm (PPA) [B.S. He, X.L. Fu, and Z.K. Jiang, Proximal-point algorithm using a linear proximal term, J. Optim. Theory Appl. 141 (2009), pp. 299–319], which aims at relaxing the requirement of the linear proximal term of classical PPA. In this paper, we make further contributions along the line. First, we generalize the linear PPA-based contraction method by using a nonlinear proximal term instead of the linear one. A notable superiority over traditional PPA-like methods is that the nonlinear proximal term of the proposed method may not necessarily be a gradient of any functions. In addition, the nonlinearity of the proximal term makes the new method more flexible. To avoid solving a variational inequality subproblem exactly, we then propose an inexact version of the developed method, which may be more computationally attractive in terms of requiring lower computational cost. Finally, we gainfully employ our new methods to solve linearly constrained convex minimization and variational inequality problems.     

Analytical integral algorithm applied to boundary layer effect and thin body effect in BEM for anisotropic potential problems

A new completely analytical integral algorithm is proposed and applied to the evaluation of nearly singular integrals in boundary element method (BEM) for two-dimensional anisotropic potential problems. The boundary layer effect and thin body effect are dealt with. The completely analytical integral formulas are suitable for the linear and non-isoparametric quadratic elements. The present algorithm applies the analytical formulas to treat nearly singular integrals. The potentials and fluxes at the interior points very close to boundary are evaluated. The unknown potentials and fluxes at boundary nodes for thin body problems with the thickness-to-length ratios from 1E−1 to 1E−8 are accurately calculated by the present algorithm. Numerical examples on heat conduction demonstrate that the present algorithm can effectively handle nearly singular integrals occurring in boundary layer effect and thin body effect in BEM. Furthermore, the present linear BEM is especially accurate and efficient for the numerical analysis of thin body problems.     

A Tailored Finite Point Method for a Singular Perturbation Problem on an Unbounded Domain

In this paper, we propose a tailored-finite-point method for a kind of singular perturbation problems in unbounded domains. First, we use the artificial boundary method (Han in Frontiers and Prospects of Contemporary Applied Mathematics, [2005]) to reduce the original problem to a problem on bounded computational domain. Then we propose a new approach to construct a discrete scheme for the reduced problem, where our finite point method has been tailored to some particular properties or solutions of the problem. From the numerical results, we find that our new methods can achieve very high accuracy with very coarse mesh even for very small ε. In the contrast, the traditional finite element method does not get satisfactory numerical results with the same mesh. Han was supported by the NSFC Project No. 10471073. Z. Huang was supported by the NSFC Projects No. 10301017, and 10676017, the National Basic Research Program of China under the grant 2005CB321701. R.B. Kellogg was supported by the Boole Centre for Research in Informatics at National University of Ireland, Cork and by Science Foundation Ireland under the Basic Research Grant Programme 2004 (Grants 04/BR/M0055, 04/BR/M0055s1).     

Optimization of a parameterized inexact Uzawa method for saddle point problems

For large sparse saddle point problems, Cao et al. studied a modified generalized parameterized inexact Uzawa (MGPIU) method (see [Y. Cao, M.Q. Jiang, L.Q. Yao, New choices of preconditioning matrices for generalized inexact parameterized iterative methods, J. Comput. Appl. Math. 235 (1) (2010) 263–269]). For iterative methods of this type, the choice of the relaxation parameter is crucial for the methods to achieve their best performance. In this paper, for an example of 2D Stokes equations, we derive the optimal relaxation parameter for the continuous version of the MGPIU method, by minimizing the corresponding convergence factor that is obtained using Fourier analysis. In addition, we find that the MGPIU method is mesh parameter independent, however, it depends asymptotically linearly on the viscosity ν, which suggests that the numerical methods for Stokes equations should be investigated with the presence of the viscosity ν, though it can be scaled out from the equations in advance. We use numerical experiments to validate our theoretical findings.     

Analysis of three-dimensional anisotropic heat conduction problems on thin domains using an advanced boundary element method

In this paper, an advanced boundary element method (BEM) is developed for solving three-dimensional (3D) anisotropic heat conduction problems in thin-walled structures. The troublesome nearly singular integrals, which are crucial in the applications of the BEM to thin structures, are calculated efficiently by using a nonlinear coordinate transformation method. For the test problems studied, promising BEM results with only a small number of boundary elements have been obtained when the thickness of the structure is in the orders of micro-scales (10^?6), which is sufficient for modeling most thin-walled structures as used in, for example, smart materials and thin layered coating systems. The advantages, disadvantages as well as potential applications of the proposed method, as compared with the finite element method (FEM), are also discussed.     

Dual reciprocity boundary element method in Laplace domain applied to anisotropic dynamic crack problems

In this paper the dual reciprocity boundary element method in the Laplace domain for anisotropic dynamic fracture mechanic problems is presented. Crack problems are analyzed using the subregion technique. The dynamic stress intensity factors are computed using traction singular quarter-point elements positioned at the tip of the crack. Numerical inversion from the Laplace domain to the time domain is achieved by the Durbin method. Numerical examples of dynamic stress intensity factor evaluation are considered for symmetric and non-symmetric problems. The influence of the number of Laplace parameters and internal points in the solution is investigated.     

A viscosity approximation scheme for finite mixed equilibrium problems and variational inequality problems and fixed point problems

In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of finite mixed equilibrium problems, the set of solutions of variational inequalities for two cocoercive mappings, the set of common fixed points of an infinite family of nonexpansive mappings and the set of common fixed points of a nonexpansive semigroup in Hilbert space. Then we prove a strong convergence theorem under some suitable conditions. The results obtained in this paper extend and improve many recent ones announced by many others.     

On the convergence of iterative methods for stabilized saddle point problems

The convergence of the inexact Uzawa method for stabilized saddle point problems was analysed in a recent paper by Cao, Evans and Qin. We show that this method converges under conditions weaker than those stated in their paper.     

A finite element method for a class of contact-impact problems

We present a finite element method for a class of contact-impact problems. Theoretical background and numerical implementation features are discussed. In particular, we consider the basic ideas of contact-impact, the assumptions which define the class of problems we deal with, spatial and temporal discretizations of the bodies involved, special problems concerning the contact of bodies of different dimensions, discrete impact and release conditions, and solution of the nonlinear algebraic problem. Several sample problems are presented which demonstrate the accuracy and versatility of the algorithm.     

SSOR-like methods for saddle point problems

Saddle point problems arise in a wide variety of applications in computational and engineering. The aim of this paper is to present a SSOR-like iterative method for solving the saddle point problems. Here the convergence of this method is studied and specifically, the spectral radius and the optimal relaxation parameter of the iteration matrix are also investigated. Numerical experiments show that the SSOR-like method with a proper preconditioning matrix is better than SOR-like method presented by Golub et al. [G.H. Golub, X. Wu, and J.-Y. Yuan, SOR-like methods for augmented systems, BIT 41 (2001), pp. 71–85].