A Fast Spectral Subtractional Solver for Elliptic Equations

The paper presents a fast subtractional spectral algorithm for the solution of the Poisson equation and the Helmholtz equation which does not require an extension of the original domain. It takes O(N ² log N) operations, where N is the number of collocation points in each direction. The method is based on the eigenfunction expansion of the right hand side with integration and the successive solution of the corresponding homogeneous equation using Modified Fourier Method. Both the right hand side and the boundary conditions are not assumed to have any periodicity properties. This algorithm is used as a preconditioner for the iterative solution of elliptic equations with non-constant coefficients. The procedure enjoys the following properties: fast convergence and high accuracy even when the computation employs a small number of collocation points. We also apply the basic solver to the solution of the Poisson equation in complex geometries.     

A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Differential Equations

We consider Lagrangian reduced-basis methods for single-parameter symmetric coercive elliptic partial differential equations. We show that, for a logarithmic-(quasi-)uniform distribution of sample points, the reduced–basis approximation converges exponentially to the exact solution uniformly in parameter space. Furthermore, the convergence rate depends only weakly on the continuity-coercivity ratio of the operator: thus very low-dimensional approximations yield accurate solutions even for very wide parametric ranges. Numerical tests (reported elsewhere) corroborate the theoretical predictions.     

Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations

In this paper we consider (hierarchical, La-grange)reduced basis approximation anda posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equa-tions. The essential ingredients are (primal-dual)Galer-kin projection onto a low-dimensional space associated with a smooth “parametric manifold” - dimension re-duction; efficient and effective greedy sampling meth-ods for identification of optimal and numerically stable approximations - rapid convergence;a posteriori er-ror estimation procedures - rigorous and sharp bounds for the linear-functional outputs of interest; and Offine-Online computational decomposition strategies - min-imummarginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control)and many-query (e.g., design optimization, multi-model/ scale)contexts. We present illustrative results for heat conduction and convection-diffusion,inviscid flow, and linear elasticity; outputs include transport rates, added mass,and stress intensity factors. This work was supported by DARPA/AFOSR Grants FA9550-05-1-0114 and FA-9550-07-1-0425,the Singapore-MIT Alliance,the Pappalardo MIT Mechanical Engineering Graduate Monograph Fund,and the Progetto Roberto Rocca Politecnico di Milano-MIT.We acknowledge many helpful discussions with Professor Yvon Maday of University Paris6.     

The Variable Order Fast Multipole Method for Boundary Integral Equations of the Second Kind

We discuss the variable order Fast Multipole Method (FMM) applied to piecewise constant Galerkin discretizations of boundary integral equations. In this version of the FMM low-order expansions are employed in the finest level and orders are increased in the coarser levels. Two versions will be discussed, the first version computes exact moments, the second is based on approximated moments. When applied to integral equations of the second kind, both versions retain the asymptotic error of the direct method. The complexity estimate of the first version contains a logarithmic term while the second version is O(N) where N is the number of panels.This work was supported by the NSF under contract DMS-0074553     

An Eulerian Formulation for Solving Partial Differential Equations Along a Moving Interface

In this paper we study an Eulerian formulation for solving partial differential equations (PDE) on a moving interface. A level set function is used to represent and capture the moving interface. A dual function orthogonal to the level set function defined in a neighborhood of the interface is used to represent some associated quantity on the interface and evolves according to a PDE on the moving interface. In particular we use a convection diffusion equation for surfactant concentration on an interface passively convected in an incompressible flow as a model problem. We develop a stable and efficient semi-implicit scheme to remove the stiffness caused by surface diffusion.     

An Approximate Solution for a Class of Second-Order Elliptic Variational Inequalities in Arbitrary-Form Domains

Penalty and dummy-domain methods are used to approximate second-order elliptic variational inequalities with a restriction inside a domain by nonlinear boundary-value problems in a rectangle. Difference schemes, with the order of accuracy O(h ^1/2) in the grid norm W ₂ ¹(), are constructed for these problems.     

一种解偏微分方程逆问题的并行算法

论述一种偏微分方程逆问题的数值解法和阵列机与 PVM平台上实现的并行算法。     

A New Class of Highly Accurate Solvers for Ordinary Differential Equations

We introduce a new class of predictor-corrector schemes for the numerical solution of the Cauchy problem for non-stiff ordinary differential equations (ODEs), obtained via the decomposition of the solutions into combinations of appropriately chosen exponentials; historically, such techniques have been known as exponentially fitted methods. The proposed algorithms differ from the classical ones both in the selection of exponentials and in the design of the quadrature formulae used by the predictor-corrector process. The resulting schemes have the advantage of significantly faster convergence, given fixed lengths of predictor and corrector vectors. The performance of the approach is illustrated via a number of numerical examples. This work was partially supported by the US Department of Defense under ONR Grant #N00014-07-1-0711 and AFOSR Grants #FA9550-06-1-0197 and #FA9550-06-1-0239.     

关于一类3-3型并联机器人运动学正解问题的研究

对一类３—３型并联机器人运动学正解问题给出一种新的解法，它将解析方法与数值方法结合起来，思路清晰，方法简明。并且应用计算机代数系统Ｍａｔｈｅｍａｔｉｃａ对结果进行了验证，实验结果表明此法正确、简单、可靠，为实际应用创造了条件     

一类半正定多项式的平方和分解及其表达式的自动生成

建立了一个把半正定稀疏多项式表为多项式平方和的算法.这一算法依赖于Hilbert第17问题的一系列经典研究结果以及实闭域上量词消去的柱形代数剖分算法.该算法的机器实现为一类代数不等式可读性证明的自动生成提供了一种非常自然的途径.     

A Learning Control for a Class of Linear Time Varying Systems Using Double Differential of Error

In this paper, a new iterative learning control based on the double differential of the error is proposed for the linear time varying system having relative degree greater than one. The convergence criterion of the proposed method is proved. Furthermore, it is shown by simulations that convergence of error can be increased considerably by using our proposed controller as compared to the iterative learning controller using error or single differential of the error for the modification of the control input without increasing the learning gain.     

A numerical algorithm coupling a bifurcating indicator and a direct method for the computation of Hopf bifurcation points in fluid mechanics

This paper deals with the computation of Hopf bifurcation points in fluid mechanics. This computation is done by coupling a bifurcation indicator proposed recently (Cadou et al., 2006) [1] and a direct method (Jackson, 1987; Jepson, 1981) [2] and [3] which consists in solving an augmented system whose solutions are Hopf bifurcation points. The bifurcation indicator gives initial critical values (Reynolds number, Strouhal frequency) for the direct method iterations. Some classical numerical examples from fluid mechanics, in two dimensions, are studied to demonstrate the efficiency and the reliability of such an algorithm.     

A direct spectral domain decomposition method for the computation of rotating flows in a T-shape geometry

This paper presents a direct domain decomposition method, coupled with a Chebyshev collocation approximation, for solving the incompressible Navier-Stokes equations in the vorticity-streamfunction formulation. The method is based on the influence matrix technique used to treat the lack of vorticity boundary conditions on no-slip walls as well as to enforce the continuity conditions at the interfaces between adjacent subdomains. The multi-domain approach is proposed in order to extend the use of spectral approximations to non-rectangular geometries and singular solutions. It is applied to the computation of a four domain configuration, corresponding to a forced throughflow in a rotating channel-cavity system which is important in air cooling devices and cannot be modeled by single-domain spectral approximations.     

A direct boundary integral equation method for the numerical construction of harmonic functions in three-dimensional layered domains containing a cavity

We consider boundary value problems for the Laplace equation in three-dimensional multilayer domains composed of an infinite strip layer of finite height and a half-space containing a bounded cavity. The unknown (harmonic) function satisfies the Neumann boundary condition on the exterior boundary of the strip layer (i.e. at the bottom of the first layer), the Dirichlet, Neumann or Robin boundary condition on the boundary surface of the cavity and the corresponding transmission (matching) conditions on the interface layer boundary. We reduce this boundary value problem to a boundary integral equation over the boundary surface of the cavity by constructing Green's matrix for the corresponding transmission problem in the domain consisting of the infinite layer and the half-space (not with the cavity). This direct integral equation approach leads, for any of the above boundary conditions, to boundary integral equations with a weak singularity on the cavity. The numerical solution of this equation is realized by Wienert's [Die Numerische approximation von Randintegraloperatoren für die Helmholtzgleichung im R ³, Ph.D. thesis, University of Göttingen, Germany, 1990] method. The reduction of the problem, originally set in an unbounded three-dimensional region, to a boundary integral equation over the boundary of a bounded domain, is computationally advantageous. Numerical results are included for various boundary conditions on the boundary of the cavity, and compared against a recent indirect approach [R. Chapko, B.T. Johansson, and O. Protsyuk, On an indirect integral equation approach for stationary heat transfer in semi-infinite layered domains in R ³ with cavities, J. Numer. Appl. Math. (Kyiv) 105 (2011), pp. 4–18], and the results obtained show the efficiency and accuracy of the proposed method. In particular, exponential convergence is obtained for smooth cavities.