A semi-discrete tailored finite point method for a class of anisotropic diffusion problems

This work proposes a tailored finite point method (TFPM) for the numerical solution of an anisotropic diffusion problem, which has much smaller diffusion coefficient along one direction than the other on a rectangular domain. The paper includes analysis on the differentiability of the solution to the given problem under some compatibility conditions. It has detailed derivation for a semi-discrete TFPM for the given problem. This work also proves a uniform error estimate on the approximate solution. Numerical results show that the TFPM is accurate as well as efficient for the strongly anisotropic diffusion problem. Examples include those that do not satisfy compatibility and regularity conditions. For the incompatible problems, numerical experiments indicate that the method proposed can still offer good numerical approximations.     

A full multigrid method for the Steklov eigenvalue problem

ABSTRACT

This paper introduces a kind of parallel multigrid method for solving Steklov eigenvalue problem based on the multilevel correction method. Instead of the common costly way of directly solving the Steklov eigenvalue problem on some fine space, the new method contains some boundary value problems on a series of multilevel finite element spaces and some steps of solving Steklov eigenvalue problems on a very low dimensional space. The linear boundary value problems are solved by some multigrid iteration steps. We will prove that the computational work of this new scheme is truly optimal, the same as solving the corresponding linear boundary value problem. Besides, this multigrid scheme has a good scalability by using parallel computing technique. Some numerical experiments are presented to validate our theoretical analysis.     

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.     

Robust multigrid solvers for the biharmonic problem in isogeometric analysis

In this paper, we develop multigrid solvers for the biharmonic problem in the framework of isogeometric analysis (IgA). In this framework, one typically sets up B-splines on the unit square or cube and transforms them to the domain of interest by a global smooth geometry function. With this approach, it is feasible to set up $H^2$-conforming discretizations. We propose two multigrid methods for such a discretization, one based on Gauss–Seidel smoothing and one based on mass smoothing. We prove that both are robust in the grid size, the latter is also robust in the spline degree. Numerical experiments illustrate the convergence theory and indicate the efficiency of the proposed multigrid approaches, particularly of a hybrid approach combining both smoothers.     

A multigrid solver for Stokes control problems

Multigrid solvers for distributed optimal control problems constrained by Stokes equations are presented. The distributed velocity tracking problem is considered with Dirichlet boundary conditions. The optimality system of the control problem that results from a Lagrange multiplier framework, forms a linear system connecting the state, adjoint, and control variables. We investigate multigrid methods on staggered grids. A coarsening by a factor of three is introduced that results in a nested hierarchy of staggered grids and simplified the intergrid transfer operators. On these grids a distributive Gauss–Seidel smoothing scheme is employed. Numerical experiments are performed to validate the effectiveness and efficiency of the proposed multigrid staggered grid framework.     

Application of splitting scheme and multigrid method for TV-Stokes denoising

Based on some previous work on the connection between image restoration and fluid dynamics,we apply a two-step algorithm for image denoising.In the first step,using a splitting scheme to study a nonlinear Stokes equation,tangent vectors are obtained.In the second step,an image is restored to fit the constructed tangent directions.We apply a fixed point iteration to solve the total variation-based image denoising problem,and use algebraic multigrid method to solve the corresponding linear equations.Numerical...     

A first-order multigrid method for bound-constrained convex optimization

The aim of this paper is to design an efficient multigrid method for constrained convex optimization problems arising from discretization of some underlying infinite dimensional problems. Due to problem dependency of this approach, we only consider bound constraints with (possibly) a single equality constraint. As our aim is to target large-scale problems, we want to avoid computation of second derivatives of the objective function, thus excluding Newton-like methods. We propose a smoothing operator that only uses first-order information and study the computational efficiency of the resulting method.     

On the robustness of the dampedV-cycle of the wavelet frequency decomposition multigrid method

The dampedV-cycle of the wavelet variation of the “Frequency decomposition multigrid method” of Hackbusch [Numer. Math.56, pp. 229–245 (1989)] is considered. It is shown that the convergence speed under sufficient damping is not affected by the presence of anisotropy but still depends on the number of levels. Our analysis is based on properties of wavelet packets which are supplied and proved. Numerical approximations to the speed of convergence illustrate the theoretical results.     

A node-nested Galerkin multigrid method for metal forging simulation

A node-nested Galerkin multigrid method is developed to solve systems provided by mixed formulations of 3D metal forming problems. An algebraic approach is used with operators built on node-nested meshes made of unstructured tetrahedra. The coarse meshes are built by an automatic coarsening algorithm based on node removal and local topological remeshing techniques. This blackbox multigrid preconditioner is developed within the PETSc library. It is plugged to the FORGE3^® finite element software with frequent remeshings. The effectiveness of the resulting multigrid solver is evaluated for several large scale problems with non-linear behaviour, showing very high efficiency. In particular, the linear rate of convergence of the method is verified on various scales simulations.     

A multigrid solver for phase field simulation of microstructure evolution

This paper presents a semi-implicit numerical method for the simulation of grain growth in two dimensions with a multi-phase field model. To avoid the strong stability condition of traditional explicit methods, a first-order, semi-implicit discretisation scheme is employed, which offers a good compromise with regard to memory intensity and computational requirements. A nonlinear multigrid solver based on the Full Approximation Scheme is implemented to solve the equations resulting from this discretisation. Simulations with the multigrid solver show that the solver has grid size independent convergence properties and is faster than a standard first-order explicit solver. As such, the multigrid solver promises to be a reliable additional computational tool for the simulation of microstructural evolution. A comparison with existing alternatives remains, however, subject of further investigation. To validate the implementation, the results of specific test cases are studied.     

A general mathematical formulation for finite-difference solution of mixed-boundary-value problems of anisotropic materials

This paper presents a general mathematical model, especially suitable for finite-difference analysis of stresses and displacements of the plane elastic problems of solid mechanics. The present formulation covers the problems of anisotropic, orthotropic and isotropic materials, in which the problem is formulated in terms of a single potential function, defined in terms of the displacement components. In addition, the formulation contains a new scheme of reduction of unknowns to be solved for a particular problem. Compared to the conventional computational approaches, the present scheme gets solution of higher accuracy and in extremely shorter time. The application of the present scheme is demonstrated here through a classical problem of solid mechanics, and the results are compared with the available solutions in the literature.     

On the behavior of a block-iterative projection method for solving convex feasibility problems

The behavior of a class of block-iterative projection algorithms for solving convex feasibility problems is studied. A limit characterization theorem and a convergence criterion are proven. Ways of accelerating the computational procedures are pointed out.     

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.     

An EXCMG accelerated multiscale multigrid computation for 3D Poisson equation

Multiscale multigrid (MSMG) method is an effective computational framework for efficiently computing high accuracy solutions for elliptic partial differential equations. In the current MSMG method, compared to the CPU cost on computing sixth-order solutions by applying extrapolation and other techniques on two fourth-order solutions from different scales grids, much more CPU time is spent on computing fourth-order solutions themselves on coarse and fine grids, particularly for high-dimensional problems. Here we propose to embed extrapolation cascadic multigrid (EXCMG) method into the MSMG framework to accelerate the whole process. Numerical results on 3D Poisson equations show that the new EXCMG–MSMG method is more efficient than the existing MSMG method and the EXCMG method for sixth-order solution computation.     

自顶向下聚集型代数多重网格预条件的健壮性与参数敏感性研究

本文针对自顶向下聚集型代数多重网格预条件,进行了健壮性与参数敏感性研究。对从各向同性与各向异性偏微分方程边值问题离散所得的多种稀疏线性方程组,首先对问题规模敏感性进行了研究,并与基于强连接的经典聚集型算法进行了系统比较,发现虽然对沿不同坐标轴具有强各向异性的问题,基于坐标分割的自顶向下聚集型算法不如基于强连接的经典聚集算法,但对其它所有情形,自顶向下聚集型算法都具有明显优势,特别是在采用Jacobi光滑时,优势更加显著。之后,对最粗网格层的分割数与每次每个子图进行分割时的分割数这两个参数进行了敏感性分析,发现在采用Jacobi光滑求解五点差分离散所得的稀疏线性方程组时,自顶向下聚集型算法对这两个参数存在敏感性,虽然大部分情形下,迭代次数比较稳定,但在少量几种情形下,迭代次数明显增加。而对从九点差分离散得到的稀疏线性方程组,以及在采用Gauss-Seidel光滑的情况下,算法对这两个参数的选取不再具有敏感性,迭代次数都比较稳定。综合分析表明,自顶向下聚集型代数多重网格预条件具有较好的健壮性,特别是在采用Gauss-Seidel光滑,或采用九点差分离散时,健壮性表现更加充分。     

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 higher-order compact ADI method with monotone iterative procedure for systems of reaction-diffusion equations

This paper is concerned with an existing compact finite difference ADI method, published in the paper by Liao et al. (2002) [3], for solving systems of two-dimensional reaction-diffusion equations with nonlinear reaction terms. This method has an accuracy of fourth-order in space and second-order in time. The existence and uniqueness of its solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear reaction terms. The convergence of the finite difference solution to the continuous solution is proved. An efficient monotone iterative algorithm is presented for solving the resulting discrete system, and some techniques for the construction of upper and lower solutions are discussed. An application using a model problem gives numerical results that demonstrate the high efficiency and advantages of the method.     

三维对流扩散方程的多重网格算法研究

研究了三维对流扩散方程基于有限差分法的多重网格算法。差分格式采用一般网格步长下的二阶中心差分格式和四阶紧致差分格式,建立了与两种格式相适应的部分半粗化的多重网格算法,构造了相应的限制算子和插值算子,并与传统的等距网格下的完全粗化的多重网格算法进行了比较。数值研究结果表明,对于各向异性问题,一般网格步长下的部分半粗化多重网格算法比等距网格下的完全粗化多重网格算法具有个更高的精度和更好的收敛效率。     

Optimization of multigrid based elliptic solver for large scale simulations in the FLASH code

FLASH is a multiphysics multiscale adaptive mesh refinement (AMR) code originally designed for simulation of reactive flows often found in Astrophysics. With its wide user base and flexible applications configuration capability, FLASH has a dual task of maintaining scalability and portability in all its solvers. The scalability of fully explicit solvers in the code is tied very closely to that of the underlying mesh. Others such as the Poisson solver based on a multigrid method have more complex scaling behavior. Multigrid methods suffer from processor starvation and dominating communication costs at coarser grids with increase in the number of processors. In this paper, we propose a combination of uniform grid mesh with AMR mesh, and the merger of two different sets of solvers to overcome the scalability limitation of the Poisson solver in FLASH. The principal challenge in the proposed merger is the efficiency of the communication algorithm to map the mesh back and forth between uniform grid and AMR. We present two different parallel mapping algorithms and also discuss results from performance studies of the two implementations. Copyright © 2012 John Wiley & Sons, Ltd.     

Sampling average approximation method for a class of stochastic Nash equilibrium problems

We consider a class of stochastic Nash equilibrium problems (SNEP). Under some mild conditions, we reformulate the SNEP as a stochastic mixed complementarity problem (SMCP). We apply the well-known sample average approximation (SAA) method to solve the SMCP. We further introduce a semismooth Newton method to solve the SAA problems. The comprehensive convergence analysis is given as well. In addition, we demonstrate the proposed approach on a stochastic Nash equilibrium model in the wholesale gas–oil markets.