Substructured two-grid and multi-grid domain decomposition methods

Two-level Schwarz domain decomposition methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level preconditioner (or its corresponding smoothing iterative method), which is based on domain decomposition techniques, and a coarse correction step, which relies on a coarse space. The coarse space must properly represent the error components that the chosen one-level method is not capable to deal with. In the literature, most of the works introduced efficient coarse spaces obtained as the span of functions defined on the entire space domain of the considered PDE. Therefore, the corresponding two-level preconditioners and iterative methods are defined in volume. In this paper, we use the excellent smoothing properties of Schwarz domain decomposition methods to define, for general elliptic problems, a new class of substructured two-level methods, for which both Schwarz smoothers and coarse correction steps are defined on the interfaces (except for the application of the smoother that requires volumetric subdomain solves). This approach has several advantages. On the one hand, the required computational effort is cheaper than the one required by classical volumetric two-level methods. On the other hand, our approach does not require, like classical multi-grid methods, the explicit construction of coarse spaces, and it permits a multilevel extension, which is desirable when the high dimension of the problem or the scarce quality of the coarse space prevents the efficient numerical solution. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.

     

Stackelberg solutions for fuzzy random two-level linear programming through probability maximization with possibility

This paper considers Stackelberg solutions for decision making problems in hierarchical organizations under fuzzy random environments. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced into the formulated fuzzy random two-level linear programming problems. On the basis of the possibility and necessity measures that each objective function fulfills the corresponding fuzzy goal, together with the introduction of probability maximization criterion in stochastic programming, we propose new two-level fuzzy random decision making models which maximize the probabilities that the degrees of possibility and necessity are greater than or equal to certain values. Through the proposed models, it is shown that the original two-level linear programming problems with fuzzy random variables can be transformed into deterministic two-level linear fractional programming problems. For the transformed problems, extended concepts of Stackelberg solutions are defined and computational methods are also presented. A numerical example is provided to illustrate the proposed methods.     

A multilevel decoupled method for a mixed Stokes/Darcy model

This paper studies decoupled numerical methods for a mixed Stokes/Darcy model for coupling fluid and porous media flows. A two-level algorithm is proposed and analyzed in Mu and Xu (2007) [10]. We generalize the two-level algorithm to a multilevel algorithm in this paper and present numerical analysis on the error estimates for the multilevel algorithm. The multilevel algorithm solves the mixed Stokes/Darcy system by applying efficient legacy code for single model solvers to solve two decoupled Stokes and Darcy subproblems on all the subsequently refined meshes, except for a much smaller global problem only on a very coarse initial mesh. Numerical experiments are conducted for both the two-level and multilevel algorithms to illustrate their effectiveness and efficiency, and validate the related theoretical analysis.     

Two-level defect-correction Oseen iterative stabilized finite element methods for the stationary Navier–Stokes equations

Two-level defect-correction Oseen iterative stabilized finite element methods for the stationary Navier–Stokes equations based on local Gauss integration are considered in this paper. The methods combine the defect-correction method and the two-level strategy with the locally stabilized method. Moreover, the stability and convergence of the presented methods are deduced. Finally, numerical tests confirm the theoretical results of the presented methods.     

非匹配网格上Stokes-Darcy 问题非协调元离散的两水平和多水平方法

本文讨论了非匹配网格上Stokes-Darcy 问题的两种低阶非协调元方法, 给出了误差估计, 对耦合的非协调元离散问题, 通过粗网格求得的界面条件, 我们提出了一个解耦的两水平算法. 并且我们将两水平方法推广到多水平情形, 其只需在一个很粗的网格上解一耦合问题, 然后在逐步加细的网格上求解解耦的问题, 理论分析和数值试验都说明方法的高效性.     

Robust semi-coarsening multilevel preconditioning of biquadratic FEM systems

While a large amount of papers are dealing with robust multilevel methods and algorithms for linear FEM elliptic systems, the related higher order FEM problems are much less studied. Moreover, we know that the standard hierarchical basis two-level splittings deteriorate for strongly anisotropic problems. A first robust multilevel preconditioner for higher order FEM systems obtained after discretizations of elliptic problems with an anisotropic diffusion tensor is presented in this paper. We study the behavior of the constant in the strengthened CBS inequality for semi-coarsening mesh refinement which is a quality measure for hierarchical two-level splittings of the considered biquadratic FEM stiffness matrices. The presented new theoretical estimates are confirmed by numerically computed CBS constants for a rich set of parameters (coarsening factor and anisotropy ratio). In the paper we consider also the problem of solving efficiently systems with the pivot block matrices arising in the hierarchical basis two-level splittings. Combining the proven uniform estimates with the theory of the Algebraic MultiLevel Iteration (AMLI) methods we obtain an optimal order multilevel algorithm whose total computational cost is proportional to the size of the discrete problem with a proportionality constant independent of the anisotropy ratio.     

混合有限元非嵌套网格型预处理法

1 引言 众所周知,二阶椭圆型问题混合有限元离散以后的矩阵是不定的,所以对混合法很难形成一种有效的区域分解法,在文[9]、[10]、[11]中提出了一些混合有限元方法的区域分解法,但在实际计算中有很多局限性。最近Chen对混合有限元法提出一种全新的解释并把它应用到多重网格法中,他的基本思想是混合有限元离散的代数系统实际上等价于某个非协调有限元离散的代数系统,这样可把一个不定问题转化为一个正定问题,本文将基于这种思想考虑混合有限元的区域分解法。 若按传统的Dryia-widlund两水平加性Schwarz方法,要求两层网格间具有嵌套关系,这样在应用中将带来很大的不便。本文将不要求粗网格嵌入细网格中,减少两层网格间的     

A two-level nesting smoothed meshfree method for structural dynamic analysis

This paper presents a novel two-level nesting smoothed meshfree method (NSMM), which significantly improves the computational efficiency of the meshfree Galerkin methods without losing their accuracy, thus facilitates the employment of meshfree methods in applications where background integration cells would be prohibitively expensive. In the NSMM, the system stiffness matrix is calculated using the general smoothing strain technique over the two-level nesting smoothing sub-domains where fewer integration points are used and the costly derivative computation of meshfree shape functions is avoided. The accuracy, efficiency and stability of the present method are assessed by virtue of several numerical examples for problems involving free and forced vibration analysis of the linear elastic continua and dynamic crack response of elastic solid. The results reveal that the NSMM stands out and achieves better performance compared to other existing approaches in the literature.     

Two-level optimization with approximate solutions in the lower level

If we want to apply iterative solution procedures of nonlinear optimization for solving the upper level of a two-level optimization problem, at each step the required problem data must be generated by solving the lower level for the actual parameter value. For the class of gradient-type methods we discuss some ideas, how the accuracy in the lower level can be controlled to ensure the convergence in the upper level. The present paper supplements results of a book of Gol'stein and Tretyakov from 1989.     

One- and two-level Schwarz methods for variational inequalities of the second kind and their application to frictional contact

We present and analyze subspace correction methods for the solution of variational inequalities of the second kind and apply these theoretical results to non smooth contact problems in linear elasticity with Tresca and non-local Coulomb friction. We introduce these methods in a reflexive Banach space, prove that they are globally convergent and give error estimates. In the context of finite element discretizations, where our methods turn out to be one- and two-level Schwarz methods, we specify their convergence rate and its dependence on the discretization parameters and conclude that our methods converge optimally. Transferring this results to frictional contact problems, we thus can overcome the mesh dependence of some fixed-point schemes which are commonly employed for contact problems with Coulomb friction.     

Convergence of multi-level iterative aggregation-disaggregation methods

This paper introduces an error propagation formula of a certain class of multi-level iterative aggregation-disaggregation (IAD) methods for numerical solutions of stationary probability vectors of discrete finite Markov chains. The formula can be used to investigate convergence by computing the spectral radius of the error propagation matrix for specific Markov chains. Numerical experiments indicate that the same type of the formula could be used for a wider class of the multi-level IAD methods. Using the formula we show that for given data there is no relation between convergence of two-level and of multi-level IAD methods.     

A system of boundary integral equations for the transmission problem in acoustics

In this paper, we reduce the classical two-dimensional transmission problem in acoustic scattering to a system of coupled boundary integral equations (BIEs), and consider the weak formulation of the resulting equations. Uniqueness and existence results for the weak solution of corresponding variational equations are established. In contrast to the coupled system in Costabel and Stephan (1985) [4], we need to take into account exceptional frequencies to obtain the unique solvability. Boundary element methods (BEM) based on both the standard and a two-level fast multipole Galerkin schemes are employed to compute the solution of the variational equation. Numerical results are presented to verify the efficiency and accuracy of the numerical methods.     

Numerical approximation of the Smagorinsky model by a two-level method

A two-level method for discretizing the Smagorinsky model for the numerical simulation of turbulent flows is proposed. In the two-level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single-step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two-level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We provide an a priori error estimate for the two-level method, which yields appropriate scalings between the coarse and fine mesh-sizes (H and h, respectively), and the radius of the spatial filter used in the Smagorinsky model (δ). In addition, we provide an algorithm in which a coarse mesh correction is performed to further enhance the accuracy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A two-level finite element method for the Navier–Stokes equations based on a new projection

We consider a fully discrete two-level approximation for the time-dependent Navier–Stokes equations in two dimension based on a time-dependent projection. By defining this new projection, the iteration between large and small eddy components can be reflected by its associated space splitting. Hence, we can get a weakly coupled system of large and small eddy components. This two-level method applies the finite element method in space and Crank–Nicolson scheme in time. Moreover,the analysis and some numerical examples are shown that the proposed two-level scheme can reach the same accuracy as the classical one-level Crank–Nicolson method with a very fine mesh size h by choosing a proper coarse mesh size H. However, the two-level method will involve much less work.     

Connections Between the Resolutions of General Two-level Factorial Designs

Indicator functions are new tools for studying two-level fractional factorial designs. This article discusses some properties of indicator functions. Using indicator functions, we study the connection between general two-level factorial designs of generalized resolutions.     

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 oper-ator. 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 perfor-mance of the methods with respect to the convergence factor, CPU-time(for one V-cycle and the setup phase) and computational complexity.     

A Class of Combined Relaxation Methods for Decomposable Variational Inequalities

Combined relaxation methods are convergent to a solution of variational inequality problems under rather mild assumptions and admit various auxiliary procedures within their two-level structure. In this work, we consider ways to construct decomposition schemes within one class of combined relaxation methods, which maintain useful convergence properties. An application to primal-dual variational inequality problems is also given.     

A two-level method for sparse time-frequency representation of multiscale data Dedicated to Professor LI TaTsien on the Occasion of His 80th Birthday

Based on the recently developed data-driven time-frequency analysis(Hou and Shi, 2013), we propose a two-level method to look for the sparse time-frequency decomposition of multiscale data. In the two-level method, we first run a local algorithm to get a good approximation of the instantaneous frequency. We then pass this instantaneous frequency to the global algorithm to get an accurate global intrinsic mode function(IMF)and instantaneous frequency. The two-level method alleviates the difficulty of the mode mixing to some extent.We also present a method to reduce the end effects.     

Stackelberg solutions for fuzzy random two-level linear programming through level sets and fractile criterion optimization

This paper considers Stackelberg solutions for two-level linear programming problems under fuzzy random environments. To deal with the formulated fuzzy random two-level linear programming problem, an α-stochastic two-level linear programming problem is defined through the introduction of α-level sets of fuzzy random variables. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced and the α-stochastic two-level linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. Through fractile criterion optimization in stochastic programming, the transformed stochastic two-level programming problem can be reduced to a deterministic two-level programming problem. An extended concept of Stackelberg solution is introduced and a numerical example is provided to illustrate the proposed method.     

Convergence of multi-level iterative aggregation–disaggregation methods

This paper introduces an error propagation formula of a certain class of multi-level iterative aggregation–disaggregation (IAD) methods for numerical solutions of stationary probability vectors of discrete finite Markov chains. The formula can be used to investigate convergence by computing the spectral radius of the error propagation matrix for specific Markov chains. Numerical experiments indicate that the same type of the formula could be used for a wider class of the multi-level IAD methods. Using the formula we show that for given data there is no relation between convergence of two-level and of multi-level IAD methods.