Two‐grid methods for time‐harmonic Maxwell equations

In this paper, we develop several two‐grid methods for the Nédélec edge finite element approximation of the time‐harmonic Maxwell equations. We first present a two‐grid method that uses a coarse space to solve the original problem and then use a fine space to solve a corresponding symmetric positive definite problem. Then, we present two types of iterative two‐grid methods, one is to add the kernel of the curl ‐operator in the fine space to a coarse mesh space to solve the original problem and the other is to use an inner iterative method for dealing with the kernel of the curl ‐operator in the fine space and the coarse space, separately. We provide the error estimates for the first two methods and present numerical experiments to show the efficiency of our methods.Copyright © 2012 John Wiley & Sons, Ltd.     

Low‐rank approximation of tensors via sparse optimization

The goal of this paper is to find a low‐rank approximation for a given nth tensor. Specifically, we give a computable strategy on calculating the rank of a given tensor, based on approximating the solution to an NP‐hard problem. In this paper, we formulate a sparse optimization problem via an l₁‐regularization to find a low‐rank approximation of tensors. To solve this sparse optimization problem, we propose a rescaling algorithm of the proximal alternating minimization and study the theoretical convergence of this algorithm. Furthermore, we discuss the probabilistic consistency of the sparsity result and suggest a way to choose the regularization parameter for practical computation. In the simulation experiments, the performance of our algorithm supports that our method provides an efficient estimate on the number of rank‐one tensor components in a given tensor. Moreover, this algorithm is also applied to surveillance videos for low‐rank approximation.     

An optimal family of exponentially accurate one‐bit Sigma‐Delta quantization schemes

Sigma‐delta modulation is a popular method for analog‐to‐digital conversion of bandlimited signals that employs coarse quantization coupled with oversampling. The standard mathematical model for the error analysis of the method measures the performance of a given scheme by the rate at which the associated reconstruction error decays as a function of the oversampling ratio λ. It was recently shown that exponential accuracy of the form O(2^−rλ) can be achieved by appropriate one‐bit sigma‐delta modulation schemes. By general information‐entropy arguments, r must be less than 1. The current best‐known value for r is approximately 0:088. The schemes that were designed to achieve this accuracy employ the “greedy” quantization rule coupled with feedback filters that fall into a class we call “minimally supported.” In this paper, we study the discrete minimization problem that corresponds to optimizing the error decay rate for this class of feedback filters. We solve a relaxed version of this problem exactly and provide explicit asymptotics of the solutions. From these relaxed solutions, we find asymptotically optimal solutions of the original problem, which improve the best‐known exponential error decay rate to r ≈ 0.102. Our method draws from the theory of orthogonal polynomials; in particular, it relates the optimal filters to the zero sets of Chebyshev polynomials of the second kind. © 2011 Wiley Periodicals, Inc.     

Web‐spline‐based finite element approximation of some quasi‐newtonian flows: Existence‐uniqueness and error bound

This article deals with the web‐spline‐based finite element approximation of quasi‐Newtonian flows. First, we consider the scalar elliptic p‐Laplace problem. Then, we consider quasi‐Newtonian flows where viscosity obeys power law or Carreau law. We prove well‐posedness at the continuous as well as the discrete level. We give some error bounds for the solution of quasi‐Newtonian flow problem based on the web‐spline method. Finally, we provide the numerical results for the p‐Laplace problem. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq31: 54–77, 2015     

Algebraic formulations for the solution of the nullspace‐free eigenvalue problem using the inexact Shift‐and‐Invert Lanczos method

Given the generalized symmetric eigenvalue problem Ax=λMx, with A semidefinite and M definite, we analyse some algebraic formulations for the approximation of the smallest non‐zero eigenpairs, assuming that a sparse basis for the null space is available. In particular, we consider the inexact version of the Shift‐and‐Invert Lanczos method, and we show that apparently different algebraic formulations provide the same approximation iterates, under some natural hypotheses. Our results suggest that alternative strategies need to be explored to really take advantage of the special problem setting, other than reformulating the algebraic problem. Experiments on a real application problem corroborate our theoretical findings. Copyright © 2002 John Wiley & Sons, Ltd.     

High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

In this study, new high‐order backward semi‐Lagrangian methods are developed to solve nonlinear advection–diffusion type problems, which are realized using high‐order characteristic‐tracking strategies. The proposed characteristic‐tracking strategies are second‐order L‐stable and third‐order L(α)‐stable methods, which are based on a classical implicit multistep method combined with a error‐correction method. We also use backward differentiation formulas and the fourth‐order finite‐difference scheme for diffusion problem discretization in the temporal and spatial domains, respectively. To demonstrate the adaptability and efficiency of these time‐discretization strategies, we apply these methods to nonlinear advection–diffusion type problems such as the viscous Burgers' equation. Through simulations, not only the temporal and spatial accuracies are numerically evaluated but also the proposed methods are shown to be superior to the compared existing characteristic‐tracking methods under the same rates of convergence in terms of accuracy and efficiency. Finally, we have shown that the proposed method well preserves the energy and mass when the viscosity coefficient becomes zero.     

Clique‐coloring some classes of odd‐hole‐free graphs

We consider the problem of clique‐coloring, that is coloring the vertices of a given graph such that no maximal clique of size at least 2 is monocolored. Whereas we do not know any odd‐hole‐free graph that is not 3‐clique‐colorable, the existence of a constant C such that any perfect graph is C‐clique‐colorable is an open problem. In this paper we solve this problem for some subclasses of odd‐hole‐free graphs: those that are diamond‐free and those that are bull‐free. We also prove the NP‐completeness of 2‐clique‐coloring K₄‐free perfect graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 233–249, 2006     

One‐level and two‐level finite element methods for the Boussinesq problem

Previous works on the convergence of numerical methods for the Boussinesq problem were conducted, while the optimal L²‐norm error estimates for the velocity and temperature are still lacked. In this paper, the backward Euler scheme is used to discrete the time terms, standard Galerkin finite element method is adopted to approximate the variables. The MINI element is used to approximate the velocity and pressure, the temperature field is simulated by the linear polynomial. Under some restriction on the time step, we firstly present the optimal L² error estimates of approximate solutions. Secondly, two‐level method based on Stokes iteration for the Boussinesq problem is developed and the corresponding convergence results are presented. By this method, the original problem is decoupled into two small linear subproblems. Compared with the standard Galerkin method, the two‐level method not only keeps good accuracy but also saves a lot of computational cost. Finally, some numerical examples are provided to support the established theoretical analysis.     

A structure‐preserving doubling algorithm for Lur'e equations

We introduce a numerical method for the numerical solution of the Lur'e equations, a system of matrix equations that arises, for instance, in linear‐quadratic infinite time horizon optimal control. We focus on small‐scale, dense problems. Via a Cayley transformation, the problem is transformed to the discrete‐time case, and the structural infinite eigenvalues of the associated matrix pencil are deflated. The deflated problem is associated with a symplectic pencil with several Jordan blocks of eigenvalue 1 and even size, which arise from the nontrivial Kronecker chains at infinity of the original problem. For the solution of this modified problem, we use the structure‐preserving doubling algorithm. Implementation issues such as the choice of the parameter γ in the Cayley transform are discussed. The most interesting feature of this method, with respect to the competing approaches, is the absence of arbitrary rank decisions, which may be ill‐posed and numerically troublesome. The numerical examples presented confirm the effectiveness of this method. Copyright © 2015 John Wiley & Sons, Ltd.     

Ritz–Galerkin method for solving a parabolic equation with non‐local and time‐dependent boundary conditions

The paper is devoted to the investigation of a parabolic partial differential equation with non‐local and time‐dependent boundary conditions arising from ductal carcinoma in situ model. Approximation solution of the present problem is implemented by the Ritz–Galerkin method, which is a first attempt at tackling parabolic equation with such non‐classical boundary conditions. In the process of dealing with the difficulty caused by integral term in non‐local boundary condition, we use a trick of introducing the transition function G(x,t) to convert non‐local boundary to another non‐classical boundary, which can be handled with the Ritz–Galerkin method. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2015 John Wiley & Sons, Ltd.     

On the Ambrosetti‐Prodi problem for a system involving p‐Laplacian operator

In this work we study an Ambrosetti‐Prodi type problem for an elliptic system involving p‐Laplacian operator. The sub and supersolution method and the Leray‐Schauder Degree Theory are used in order to prove our result.     

Complexity of clique‐coloring odd‐hole‐free graphs

In this paper we investigate the problem of clique‐coloring, which consists in coloring the vertices of a graph in such a way that no monochromatic maximal clique appears, and we focus on odd‐hole‐free graphs. On the one hand we do not know any odd‐hole‐free graph that is not 3‐clique‐colorable, but on the other hand it is NP‐hard to decide if they are 2‐clique‐colorable, and we do not know if there exists any bound k₀ such that they are all k₀ ‐clique‐colorable. First we will prove that (odd hole, codiamond)‐free graphs are 2‐clique‐colorable. Then we will demonstrate that the complexity of 2‐clique‐coloring odd‐hole‐free graphs is actually Σ₂ P‐complete. Finally we will study the complexity of deciding whether or not a graph and all its subgraphs are 2‐clique‐colorable. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 139–156, 2009     

Numerical analysis of a locking‐free mixed finite element method for a bending moment formulation of Reissner‐Mindlin plate model

This article deals with the approximation of the bending of a clamped plate, modeled by Reissner‐Mindlin equations. It is known that standard finite element methods applied to this model lead to wrong results when the thickness t is small. Here, we propose a mixed formulation based on the Hellinger‐Reissner principle which is written in terms of the bending moments, the shear stress, the rotations and the transverse displacement. To prove that the resulting variational formulation is well posed, we use the Babu?ka‐Brezzi theory with appropriate t ‐dependent norms. The problem is discretized by standard mixed finite elements without the need of any reduction operator. Error estimates are proved. These estimates have an optimal dependence on the mesh size h and a mild dependence on the plate thickness t. This allows us to conclude that the method is locking‐free. The proposed method yields direct approximation of the bending moments and the shear stress. A local postprocessing leading to H¹ ‐type approximations of transverse displacement and rotations is introduced. Moreover, we propose a hybridization procedure, which leads to solving a significantly smaller positive definite system. Finally, we report numerical experiments which allow us to assess the performance of the method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

The hp‐version of the BEM with quasi‐uniform meshes for a three‐dimensional crack problem: The case of a smooth crack having smooth boundary curve

The article considers a three‐dimensional crack problem in linear elasticity with Dirichlet boundary conditions. The crack in this model problem is assumed to be a smooth open surface with smooth boundary curve. The hp‐version of the boundary element method with weakly singular operator is applied to approximate the unknown jump of the traction which is not L₂‐regular due to strong edge singularities. Assuming quasi‐uniform meshes and uniform distributions of polynomial degrees, we prove an a priori error estimate in the energy norm. The estimate gives an upper bound for the error in terms of the mesh size h and the polynomial degree p. It is optimal in h for any given data and quasi‐optimal in p for sufficiently smooth data. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A new stability number of the Bresse‐Cattaneo system

In this paper, we consider the Bresse‐Cattaneo system with a frictional damping term and prove some optimal decay results for the L²‐norm of the solution and its higher order derivatives. In fact, we show that there is a completely new stability number δ that controls the decay rate of the solution. To prove our results, we use the energy method in the Fourier space to build some very delicate Lyapunov functionals that give the desired results. We also prove the optimality of the results by using the eigenvalues expansion method. In addition, we show that for the absence of the frictional damping term, the solution of our problem does not decay at all. This result improves some early results     

Two‐level discretization of the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity

The r‐Laplacian has played an important role in the development of computationally efficient models for applications, such as numerical simulation of turbulent flows. In this article, we examine two‐level finite element approximation schemes applied to the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity, where r is the order of the power‐law artificial viscosity term. 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 rigorous numerical analysis of the two‐level approximation scheme and derive scalings which vary based on the coefficient r, coarse mesh size H, fine mesh size h, and filter radius δ. We also investigate the two‐level algorithm in several computational settings, including the 3D numerical simulation of flow past a backward‐facing step at Reynolds number Re = 5100. In all numerical tests, the two‐level algorithm was proven to achieve the same order of accuracy as the standard one‐level algorithm, at a fraction of the computational cost. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A multilevel finite element method in space‐time for the Navier‐Stokes problem

A multilevel finite element method in space‐time for the two‐dimensional nonstationary Navier‐Stokes problem is considered. The method is a multi‐scale method in which the fully nonlinear Navier‐Stokes problem is only solved on a single coarsest space‐time mesh; subsequent approximations are generated on a succession of refined space‐time meshes by solving a linearized Navier‐Stokes problem about the solution on the previous level. The a priori estimates and error analysis are also presented for the J‐level finite element method. We demonstrate theoretically that for an appropriate choice of space and time mesh widths: h_j ～ h, k_j ～ k, j = 2, …, J, the J‐level finite element method in space‐time provides the same accuracy as the one‐level method in space‐time in which the fully nonlinear Navier‐Stokes problem is solved on a final finest space‐time mesh. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

An efficient numerical treatment of fourth‐order fractional diffusion‐wave problems

In this paper, we consider the numerical treatment of a fourth‐order fractional diffusion‐wave problem. Our proposed method includes the use of parametric quintic spline in the spatial dimension and the weighted shifted Grünwald‐Letnikov approximation of fractional integral. The solvability, stability, and convergence of the numerical scheme are rigorously proved. It is shown that the theoretical convergence order improves those of earlier work. Simulation is further carried out to demonstrate the numerical efficiency of the proposed scheme and to compare with other methods.     

Stability and convergence of two‐grid Crank‐Nicolson extrapolation scheme for the time‐dependent natural convection equations

In this paper, we consider the Crank‐Nicolson extrapolation scheme for the 2D/3D unsteady natural convection problem. Our numerical scheme includes the implicit Crank‐Nicolson scheme for linear terms and the recursive linear method for nonlinear terms. Standard Galerkin finite element method is used to approximate the spatial discretization. Stability and optimal error estimates are provided for the numerical solutions. Furthermore, a fully discrete two‐grid Crank‐Nicolson extrapolation scheme is developed, the corresponding stability and convergence results are derived for the approximate solutions. Comparison from aspects of the theoretical results and computational efficiency, the two‐grid Crank‐Nicolson extrapolation scheme has the same order as the one grid method for velocity and temperature in H¹‐norm and for pressure in L²‐norm. However, the two‐grid scheme involves much less work than one grid method. Finally, some numerical examples are provided to verify the established theoretical results and illustrate the performances of the developed numerical schemes.     

A Riesz–Feller space‐fractional backward diffusion problem with a time‐dependent coefficient: regularization and error estimates

In this paper, we consider a Riesz–Feller space‐fractional backward diffusion problem with a time‐dependent coefficient We show that this problem is ill‐posed; therefore, we propose a convolution regularization method to solve it. New error estimates for the regularized solution are given under a priori and a posteriori parameter choice rules, respectively. Copyright © 2016 John Wiley & Sons, Ltd.