Strengthened Cauchy–Bunyakowski–Schwarz inequality for a three‐dimensional elasticity system

The constant γ of the strengthened Cauchy–Bunyakowski–Schwarz (CBS) inequality plays a fundamental role in the convergence rate of multilevel iterative methods. The main purpose of this work is to give an estimate of the constant γ for a three‐dimensional elasticity system. The theoretical results obtained are practically important for the successful implementation of the finite element method to large‐scale modelling of complicated structures as they allow us to construct optimal order algebraic multilevel iterative solvers for a wide class of real‐life elasticity problems. Copyright © 2001 John Wiley & Sons, Ltd.     

A CFL‐free explicit characteristic interior penalty scheme for linear advection‐reaction equations

We develop a CFL‐free, explicit characteristic interior penalty scheme (CHIPS) for one‐dimensional first‐order advection‐reaction equations by combining a Eulerian‐Lagrangian approach with a discontinuous Galerkin framework. The CHIPS method retains the numerical advantages of the discontinuous Galerkin methods as well as characteristic methods. An optimal‐order error estimate in the L² norm for the CHIPS method is derived and numerical experiments are presented to confirm the theoretical estimates. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Sharp estimates for finite element approximations to parabolic problems with Neumann boundary data of low regularity

Consider a homogeneous parabolic problem on a smooth bounded domain in ℝ ^N but with initial data and Neumann boundary data of low regularity. Sharp interior maximum norm error estimates are given for a semidiscrete C ⁰ finite element approximation to this problem. These estimates are obtained by first establishing a new localized L ^∞ estimate for semidiscrete finite element approximations on interior subdomains. Numerical examples illustrate the findings. AMS subject classification (2000) 65N30     

Discontinuous Stable Elements for the Incompressible Flow

In this paper, we derive a discontinuous Galerkin finite element formulation for the Stokes equations and a group of stable elements associated with the formulation. We prove that these elements satisfy the new inf–sup condition and can be used to solve incompressible flow problems. Associated with these stable elements, optimal error estimates for the approximation of both velocity and pressure in L ² norm are obtained for the Stokes problems, as well as an optimal error estimate for the approximation of velocity in a mesh dependent norm.     

Adaptive finite element methods for Boussinesq equations

An a posteriori error analysis for Boussinesq equations is derived in this article. Then we compare this new estimate with a previous one developed for a regularized version of Boussinesq equations in a previous work. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 214–236, 2000     

A direct method for the general solution of a system of linear equations

A computationally stable method for the general solution of a system of linear equations is given. The system isA ^Tx–B=0, where then-vectorx is unknown and then×q matrixA and theq-vectorB are known. It is assumed that the matrixA ^T and the augmented matrix [A ^T,B] are of the same rankm, wheremn, so that the system is consistent and solvable. Whenm<n, the method yields the minimum modulus solutionx _m and a symmetricn ×n matrixH _m of rankn–m, so thatx=x _m+H _my satisfies the system for ally, ann-vector. Whenm=n, the matrixH _m reduces to zero andx _m becomes the unique solution of the system.The method is also suitable for the solution of a determined system ofn linear equations. When then×n coefficient matrix is ill-conditioned, the method can produce a good solution, while the commonly used elimination method fails.This research was supported by the National Science Foundation, Grant No. GP-41158.     

A uniformly optimal‐order estimate for finite volume method for advection‐diffusion equations

We prove an optimal‐order error estimate in a weighted energy norm for finite volume method for two‐dimensional time‐dependent advection–diffusion equations on a uniform space‐time partition of the domain. The generic constants in the estimates depend only on certain norms of the true solution but not on the scaling parameter. These estimates, combined with a priori stability estimates of the governing partial differential equations with full regularity, yield a uniform estimate of the finite volume method, in which the generic constants depend only on the Sobolev norms of the initial and right side data but not on the scaling parameter. We use the interpolation of spaces and stability estimates to derive a uniform estimate for problems with minimal or intermediate regularity, where the convergence rates are proportional to certain Besov norms of the initial and right‐hand side data. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 17‐43, 2014     

Why Gaussian quadrature in the complex plane?

This paper synthesizes formally orthogonal polynomials, Gaussian quadrature in the complex plane and the bi-conjugate gradient method together with an application. Classical Gaussian quadrature approximates an integral over (a region of) the real line. We present an extension of Gaussian quadrature over an arc in the complex plane, which we call complex Gaussian quadrature. Since there has not been any particular interest in the numerical evaluation of integrals over the long history of complex function theory, complex Gaussian quadrature is in need of motivation. Gaussian quadrature in the complex plane yields approximations of certain sums connected with the bi-conjugate gradient method. The scattering amplitude c ^T A ^–1 b is an example where A is a discretization of a differential–integral operator corresponding to the scattering problem and b and c are given vectors. The usual method to estimate this is to use c ^T x ^(k). A result of Warnick is that this is identically equal to the complex Gaussian quadrature estimate of 1/. Complex Gaussian quadrature thereby replaces this particular inner product in the estimate of the scattering amplitude.     

Long-step strategies in interior-point primal-dual methods

In this paper we analyze from a unique point of view the behavior of path-following and primal-dual potential reduction methods on nonlinear conic problems. We demonstrate that most interior-point methods with efficiency estimate can be considered as different strategies of minimizing aconvex primal-dual potential function in an extended primal-dual space. Their efficiency estimate is a direct consequence of large local norm of the gradient of the potential function along a central path. It is shown that the neighborhood of this path is a region of the fastest decrease of the potential. Therefore the long-step path-following methods are, in a sense, the best potential-reduction strategies. We present three examples of such long-step strategies. We prove also an efficiency estimate for a pure primal-dual potential reduction method, which can be considered as an implementation of apenalty strategy based on a functional proximity measure. Using the convex primal dual potential, we prove efficiency estimates for Karmarkar-type and Dikin-type methods as applied to a homogeneous reformulation of the initial primal-dual problem.     

On the convergence of a class of derivative-free minimization algorithms

A convergence analysis is presented for a general class of derivative-free algorithms for minimizing a functionf(x) for which the analytic form of the gradient and the Hessian is impractical to obtain. The class of algorithms accepts finite-difference approximation to the gradient, with stepsizes chosen in such a way that the length of the stepsize must meet two conditions involving the previous stepsize and the distance from the last estimate of the solution to the current estimate. The algorithms also maintain an approximation to the second-derivative matrix and require that the change inx made at each iteration be subject to a bound that is also revised automatically. The convergence theorems have the features that the starting pointx ¹ need not be close to the true solution andf(x) need not be convex. Furthermore, despite the fact that the second-derivative approximation may not converge to the true Hessian at the solution, the rate of convergence is still Q-superlinear. The theorry is also shown to be applicable to a modification of Powell's dog-leg algorithm.     

Sequential conjugate-gradient-restoration algorithm for optimal control problems. Part 1. Theory

This paper considers the problem of minimizing a functionalI which depends on the statex(t), the controlu(t), and the parameter π. Here,I is a scalar,x ann-vector,u anm-vector, and π ap-vector. At the initial point, the state is prescribed. At the final point, the state and the parameter are required to satisfyq scalar relations. Along the interval of integration, the state, the control, and the parameter are required to satisfyn scalar differential equations. First, the case of a quadratic functional subject to linear constraints is considered, and a conjugate-gradient algorithm is derived. Nominal functionsx(t),u(t), π satisfying all the differential equations and boundary conditions are assumed. Variations Δx(t), δu(t), Δπ are determined so that the value of the functional is decreased. These variations are obtained by minimizing the first-order change of the functional subject to the differential equations, the boundary conditions, and a quadratic constraint on the variations of the control and the parameter. Next, the more general case of a nonquadratic functional subject to nonlinear constraints is considered. The algorithm derived for the linear-quadratic case is employed with one modification: a restoration phase is inserted between any two successive conjugate-gradient phases. In the restoration phase, variations Δx(t), Δu(t), Δπ are determined by requiring the least-square change of the control and the parameter subject to the linearized differential equations and the linearized boundary conditions. Thus, a sequential conjugate-gradient-restoration algorithm is constructed in such a way that the differential equations and the boundary conditions are satisfied at the end of each complete conjugate-gradient-restoration cycle. Several numerical examples illustrating the theory of this paper are given in Part 2 (see Ref. 1). These examples demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper. This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-72-2185. The authors are indebted to Professor A. Miele for stimulating discussions. Formerly, Graduate Studient in Aero-Astronautics, Department of Mechanical and Aerospace Engineering and Materials Science, Rice University, Houston, Texas.     

On the subdomain‐Galerkin/least squares method for 2‐ and 3‐D mixed elliptic problems with reaction terms

In this article we apply the subdomain‐Galerkin/least squares method, which is first proposed by Chang and Gunzburger for first‐order elliptic systems without reaction terms in the plane, to solve second‐order non‐selfadjoint elliptic problems in two‐ and three‐dimensional bounded domains with triangular or tetrahedral regular triangulations. This method can be viewed as a combination of a direct cell vertex finite volume discretization step and an algebraic least‐squares minimization step in which the pressure is approximated by piecewise linear elements and the flux by the lowest order Raviart‐Thomas space. This combined approach has the advantages of both finite volume and least‐squares methods. Among other things, the combined method is not subject to the Ladyzhenskaya‐Babus?ka‐Brezzi condition, and the resulting linear system is symmetric and positive definite. An optimal error estimate in the H¹(Ω) × H(div; Ω) norm is derived. An equivalent residual‐type a posteriori error estimator is also given. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 738–751, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10030.     

BDDC methods for discontinuous Galerkin discretization of elliptic problems

A discontinuous Galerkin (DG) discretization of Dirichlet problem for second-order elliptic equations with discontinuous coefficients in 2-D is considered. For this discretization, balancing domain decomposition with constraints (BDDC) algorithms are designed and analyzed as an additive Schwarz method (ASM). The coarse and local problems are defined using special partitions of unity and edge constraints. Under certain assumptions on the coefficients and the mesh sizes across ∂Ω_i, where the Ω_i are disjoint subregions of the original region Ω, a condition number estimate C(1+max_ilog(H_i/h_i))² is established with C independent of h_i, H_i and the jumps of the coefficients. The algorithms are well suited for parallel computations and can be straightforwardly extended to the 3-D problems. Results of numerical tests are included which confirm the theoretical results and the necessity of the imposed assumptions.     

A Parallel Algorithm for the Estimation of the Global Error in Runge–Kutta Methods

The object of this work is the estimate of the global error in the numerical solution of the IVP for a system of ODE's. Given a Runge–Kutta formula of order q, which yields an approximation y _n to the true value y(x _n ), a general, parallel method is presented, that provides a second value y _n ^* of order q+2; the global error e _n =y _n –y(x _n ) is then estimated by the difference y _n –y _n ^*. The numerical tests reported, show the very good performance of the procedure proposed. A comparison with the code GEM90 is also appended.     

Upper Bound of the Constant in Strengthened C.B.S. Inequality for Systems of Linear Partial Differential Equations

The constant in the strengthened Cauchy–Bunyakowski–Schwarz (C.B.S.) inequality plays a crucial role in the convergence rate of multilevel iterative methods as well as in the efficiency of a posteriori error estimators, that is the framework of finite element approximations of systems of partial differential equations. We consider an approximation of general systems of linear partial differential equations in R ³. Concerning a multilevel convergence rate corresponding to nested general tetrahedral meshes of size h and 2h, we give an estimate of this constant for general three-dimensional cases.     

Proximity control in bundle methods for convex nondifferentiable minimization

Proximal bundle methods for minimizing a convex functionf generate a sequence {x ^k } by takingx ^k+1 to be the minimizer of , where is a sufficiently accurate polyhedral approximation tof andu ^k > 0. The usual choice ofu ^k = 1 may yield very slow convergence. A technique is given for choosing {u ^k } adaptively that eliminates sensitivity to objective scaling. Some encouraging numerical experience is reported.This research was supported by Project CPBP.02.15.     

A unified theory for homotopy principles for multimaps

In this article, we present a definition of d-essential and d–L-essential maps in completely regular topological spaces and we establish a homotopy property for both d-essential and d–L-essential maps. Also using the notion of extendability, we present new continuation theorems.     

Error analysis of the L2 least‐squares finite element method for incompressible inviscid rotational flows

In this article we analyze the L² least‐squares finite element approximations to the incompressible inviscid rotational flow problem, which is recast into the velocity‐vorticity‐pressure formulation. The least‐squares functional is defined in terms of the sum of the squared L² norms of the residual equations over a suitable product function space. We first derive a coercivity type a priori estimate for the first‐order system problem that will play the crucial role in the error analysis. We then show that the method exhibits an optimal rate of convergence in the H¹ norm for velocity and pressure and a suboptimal rate of convergence in the L² norm for vorticity. A numerical example in two dimensions is presented, which confirms the theoretical error estimates. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

An acceleration of gradient descent algorithm with backtracking for unconstrained optimization

In this paper we introduce an acceleration of gradient descent algorithm with backtracking. The idea is to modify the steplength t _k by means of a positive parameter θ _k , in a multiplicative manner, in such a way to improve the behaviour of the classical gradient algorithm. It is shown that the resulting algorithm remains linear convergent, but the reduction in function value is significantly improved.     

Analysis and convergence of finite volume method using discontinuous bilinear functions

We develop finite volume method using discontinuous bilinear functions on rectangular mesh. This method is analyzed for the Stokes equations. An optimal error estimate for the approximation of velocity is obtained in a mesh‐dependent norm. First order L²‐error estimates are derived for the approximations of both velocity and pressure. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007