Pointwise Sense Research Articles

Higher-order multi-scale computational approach and its convergence for nonlocal gradient elasticity problems of composite materials

A novel higher-order multi-scale computational approach for efficient nonlocal gradient elasticity simulation of composite materials is proposed in this paper. Based on the use of decoupling technique, raw fourth-order nonlocal gradient equations with periodically oscillatory coefficients are decoupled as the new second-order multi-scale equations for reducing the continuity requirements when implementing multi-scale simulation. Next, a novel macro-micro coupled multi-scale computational model with higher-order corrected terms is rigorously devised for high-resolution multi-scale and nonlocal analysis of composite materials via multi-scale asymptotic analysis. Then, the local error analysis of higher-order multi-scale solutions in the point-wise sense illustrates that establishing higher-order asymptotic corrected terms plays a vital role in investigating the micromechanical mechanisms. Moreover, a rigorous global error estimation with an explicit rate of higher-order multi-scale solutions is first derived in the energy norm sense. In addition, an efficient multi-scale numerical algorithm is presented to effectively simulate nonlocal gradient elasticity problems of composite materials based on finite element method (FEM). And we also derive the convergence estimation of the proposed numerical algorithm. Finally, numerical experiments are conducted to gauge the efficiency and accuracy of the proposed higher-order multi-scale method. This paper offers a unified higher-order two-scale computational framework for enabling nonlocal gradient elasticity behavior simulation of composite materials.

Pointwise error estimate of conservative difference scheme for supergeneralized viscous Burgers' equation

<abstract><p>This work focuses on exploring pointwise error estimate of three-level conservative difference scheme for supergeneralized viscous Burgers' equation. The cut-off function method plays an important role in constructing difference scheme and presenting numerical analysis. We study the conservative invariant of proposed method, which is energy-preserving for all positive integers $ p $ and $ q $. Meanwhile, one could apply the discrete energy argument to the rigorous proof that the three-level scheme has unique solution combining the mathematical induction. In addition, we prove the $ L_2 $-norm and $ L_{\infty} $-norm convergence of proposed scheme in pointwise sense with separate and different ways, which is different from previous work in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. Numerical results verify the theoretical conclusions.</p></abstract>

Pointwise superconvergence of block finite elements for the three-dimensional variable coefficient elliptic equation

<p>This study investigated the point-wise superconvergence of block finite elements for the variable coefficient elliptic equation in a regular family of rectangular partitions of the domain in three-dimensional space. Initially, the estimates for the three-dimensional discrete Greens function and discrete derivative Greens function were presented. Subsequently, employing an interpolation operator of projection type, two essential weak estimates were derived, which were crucial for superconvergence analysis. Ultimately, by combining the aforementioned estimates, we achieved superconvergence estimates for the derivatives and function values of the finite element approximation in the point-wise sense of the $ L^\infty $-norm. A numerical example illustrated the theoretical results.</p>

The virtual element method for a 2D incompressible MHD system

We present a novel discretization for the two-dimensional incompressible Magnetohydrodynamics (MHD) system coupling an electromagnetic model and a fluid flow model. Our approach follows the framework of the Virtual Element Method and offers two main advantages. The method can be implemented on unstructured meshes making it highly versatile and capable of handling a broad set of problems involving interfaces, free-boundaries, or adaptive refinements of the mesh. The second advantage concerns the divergence of the magnetic flux field and the fluid velocity. Our approach guarantees that the numerical approximation of the magnetic flux field and the fluid velocity are divergence free if their initial states are divergence free. Importantly, the divergence-free condition for the fluid velocity is satisfied in a pointwise sense. We include a theoretical proof of the condition on the magnetic flux field, energy estimates and a well-posedness study. Numerical testing confirms robustness of the method and its convergence properties on a variety of meshes.

Onsager’s Energy Conservation of Weak Solutions for a Compressible and Inviscid Fluid

In this article, two classes of sufficient conditions of weak solutions are given to guarantee the energy conservation of the compressible Euler equations. Our strategy is to introduce a test function φ(t)vϵ to derive the total energy. The velocity field v needs to be regularized both in time and space. In contrast to the noncompressible Euler equations, the compressible flows we consider here do not have a divergence-free structure. Therefore, it is necessary to make an additional estimate of the pressure p, which takes advantage of an appropriate commutator. In addition, by using the weak convergence, we show that the energy equality is conserved in a point-wise sense.

$ W^{1, \infty} $-seminorm superconvergence of the block finite element method for the five-dimensional Poisson equation

<abstract><p>This study focused on the superconvergence of the finite element method for the five-dimensional Poisson equation in the $ W^{1, \infty} $-seminorm. Specifically, we investigated the block finite element method, which is a tensor-product finite element approach applied to regular rectangular partitions of the domain. First, we introduced the finite element scheme for the equation and discussed various functions related to it, along with their properties. Next, we proposed a weight function and established its important properties, which play a crucial role in the theoretical analysis. By utilizing the properties of the weight function and employing weighted-norm analysis techniques, we derived an optimal order estimate in the $ W^{2, 1} $-seminorm for the discrete derivative Green's function (DDGF). Furthermore, we provided an interpolation fundamental estimate of the second type, also known as the weak estimate of the second type, for the block finite element. This weak estimate is based on a five-dimensional interpolation operator of the projection type. Finally, by combining the derived $ W^{2, 1} $-seminorm estimate for the DDGF and the weak estimate for the block finite element, we obtained a superconvergence estimate for the block finite element approximation in the pointwise sense of the $ W^{1, \infty} $-seminorm. The correctness of the theoretical results was demonstrated through a numerical example.</p></abstract>

Scalar curvature lower bound under integral convergence

In this work, we consider sequences of \(C^2\) metrics which converges to a \(C^2\) metric in \(C^0\) sense. We show that if the scalar curvature of the sequence is almost non-negative in the integral sense, then the limiting metric has scalar curvature lower bound in point-wise sense.

Multi-scale computational method for nonlinear dynamic thermo-mechanical problems of composite materials with temperature-dependent properties

For most of materials, the properties of the materials are usually dependent on the temperature, which causes the nonlinear physical and mechanical material behaviors. In this paper, an innovative multi-scale computational method is developed for efficiently simulating nonlinear dynamic thermo-mechanical problems of composite materials. The nonlinearities of these multi-scale problems were caused by the temperature-dependent properties of each component material in the composites and the heterogeneities of composite materials are taken into account by periodic distributions of representative unit cells on the micro-scale. Firstly, a second-order two-scale (SOTS) computational model for accurately analyzing nonlinear dynamic thermo-mechanical behaviors of composite materials is established based on multi-scale asymptotic analysis and Taylor series method. The proposed SOTS computational model includes the first-order and second-order auxiliary cell problems on micro-scale, homogenized problem on macro-scale and macro–micro coupled SOTS asymptotic solutions. Then we theoretically explain the crucial importance of developing the SOTS solutions by the error analysis in the point-wise sense. Next, a rigorous error analysis with an explicit rate of the SOTS approximate solutions is given under some simplifications and assumptions in the integral sense. In addition, a multi-scale numerical algorithm is proposed in detail to effectively simulate these nonlinear multi-scale problems. Finally, several typical numerical examples are carried out to confirm the effectiveness and correctness of the proposed multi-scale numerical algorithm, especially for large-scale engineering structures. This study offers an efficient and high-accuracy multi-scale computational scheme that enables the effective simulation and analysis of nonlinear dynamic thermo-mechanical problems of composite materials with temperature-dependent properties.

An inexact ADMM with proximal-indefinite term and larger stepsize

In this paper, an inexact Alternating Direction Method of Multipliers (ADMM) has been proposed for solving the two-block separable convex optimization problem subject to linear equality constraints. The first resulting subproblem is solved inexactly under relative error criterion, while another subproblem called regularization problem is solved inexactly by introducing an indefinite proximal term. Meanwhile, the dual variable is updated twice with relatively larger stepsizes since an indefinite proximal term is exploited. By reformulating the first-order optimality conditions of the involved subproblems as a monotone inclusion problem, the sublinear convergence rate of the proposed algorithm is established in the pointwise and ergodic sense. Numerical experiments on testing two sparse optimization problems from statistical learning and image restoration indicate that our inexact ADMM performs much better than several well-established algorithms.

The analysis and computation on nonlocal thermoelastic problems of blend composites via enriched second-order multi-scale computational method

This paper proposes an innovative enriched second-order multi-scale (SOMS) computational method to simulate and analyze the nonlocal thermoelastic problems of blend composites with stress and heat flux gradient behaviors. The multiple periodical heterogeneities and periodic configurations of investigated blend composites in different substructures result in a huge computational cost for direct numerical simulations. The significant characteristics of this study are as follows. (1) The nonlocal properties of blend composites in constitutive equations are converted into the source terms of thermoelastic balance equations. The novel macro-micro coupled SOMS computational model for these transformed nonlocal multi-scale problems is derived on the basis of multi-scale asymptotic analysis. The nonlocal thermoelastic behaviors of blend composites can be merely uncovered in the enriched SOMS solutions. (2) The error analysis in the pointwise sense is presented to elucidate the importance and necessity of establishing the enriched SOMS solutions. Furthermore, an explicit error estimate for the SOMS approximate solutions is obtained in the integral sense for these nonlocal multi-scale problems. (3) A multi-scale numerical algorithm is presented to effectively simulate nonlocal thermoelastic problems of blend composites based on finite element method (FEM). Finally, the capability of the proposed enriched SOMS computational method is demonstrated by typical two-dimensional (2D) and three-dimensional (3D) blend composites, presenting not only the excellent numerical accuracy but also the less computational cost. This work proposes a unified multi-scale computational framework for enabling nonlocal thermoelastic behavior analysis of blend composites.

Global existence and finite time blow-up for a stochastic non-local reaction-diffusion equation

In this paper, we investigate the existence and blow-up of positive solutions for a stochastic non-local reaction-diffusion equation. Firstly, we give the conditions for the existence of global positive solutions and estimate the probability of existence of non-trivial positive global solution. Secondly, by choosing some special initial data, we prove that stochastic non-local reaction-diffusion equation blows up in finite time in the point-wise sense with probability 1. The random upper bounds for blow-up times are obtained. Furthermore, by increasing the amplitude of the initial data, we can get a blow-up in any short time with a positive probability. Finally, we prove that the blow-up set is the whole region.

Computationally efficient higher-order three-scale method for nonlocal gradient elasticity problems of heterogeneous structures with multiple spatial scales

In this work, an innovative higher-order three-scale (HOTS) computational approach is developed, which can handle and simulate the nonlocal strain-stress gradient elasticity model of heterogeneous structures with multiple spatial scales. The significant characteristics of this study are: (i) the raw fourth-order nonlocal gradient elasticity equations with three-scale structural characteristics are decoupled as the new multi-scale second-order equations for reducing the continuity requirements for multi-scale computational method. (ii) a new micro-meso-macro coupled HOTS computational model for accurately analyzing the nonlocal gradient elasticity behaviors of heterogeneous structures with three-scale structural configurations is rigorously devised by virtue of multi-scale asymptotic analysis. (iii) the small-scale approximate performance of HOTS solutions is derived in the pointwise sense, that illustrates the crucial indispensability of establishing the HOTS solutions for capturing the micro-scale, meso-scale and macro-scale gradient elasticity behaviors of the heterogeneous structures accurately. (iv) the efficient three-scale numerical algorithm on the basis of finite element method (FEM) is presented at length. Finally, several numerical experiments are carried out to validate the applicability of the established HOTS computational methodology, presenting not only the excellent numerical accuracy, but also the advanced computational efficiency. This work provides a unified three-scale computational framework which enables the accurate analysis and efficient simulation of nonlocal strain-stress gradient elasticity problems of heterogeneous structures with multiple spatial scales, and provides a potential application for practical engineering computation.

Integral operators defined “up to a polynomial”

We introduce a suitable notion of integral operators (comprising the fractional Laplacian as a particular case) acting on functions with minimal requirements at infinity. For these functions, the classical definition would lead to divergent expressions, thus we replace it with an appropriate framework obtained by a cut-off procedure. The notion obtained in this way quotients out the polynomials which produce the divergent pattern once the cut-off is removed. We also present results of stability under the appropriate notion of convergence and compatibility results between polynomials of different orders. Additionally, we address the solvability of the Dirichlet problem. The theory is developed in general in the pointwise sense. A viscosity counterpart is also presented under the additional assumption that the interaction kernel has a sign, in conformity with the maximum principle structure.

Numerical analysis of a fourth-order linearized difference method for nonlinear time-space fractional Ginzburg-Landau equation

<abstract><p>An efficient difference method is constructed for solving one-dimensional nonlinear time-space fractional Ginzburg-Landau equation. The discrete method is developed by adopting the $ L2 $-$ 1_{\sigma} $ scheme to handle Caputo fractional derivative, while a fourth-order difference method is invoked for space discretization. The well-posedness and a priori bound of the numerical solution are rigorously studied, and we prove that the difference scheme is unconditionally convergent in pointwise sense with the rate of $ O(\tau^2+h^4) $, where $ \tau $ and $ h $ are the time and space steps respectively. In addition, the proposed method is extended to solve two-dimensional problem, and corresponding theoretical analysis is established. Several numerical tests are also provided to validate our theoretical analysis.</p></abstract>

Pointwise error estimate and stability analysis of fourth-order compact difference scheme for time-fractional Burgers’ equation

A novel implicit high-order compact difference scheme is established for the time-fractional Burgers’ equation based on a newly developed nonlinear compact operator and the reduction order technique under the uniform mesh and graded mesh. Uniqueness, boundedness, convergence in the pointwise sense and stability in L2-norm of the discrete solutions are proved by the Browder fixed point theorem and energy argument. Numerical examples with the smooth solution and nonsmooth solution are provided to confirm numerical theoretical results and demonstrate the efficiency of the high-order compact difference scheme.

S, p-Harmonic Approximation of Functions of LeastWs,l-Seminorm

AbstractWe investigate the convergence as $p\searrow 1$ of the minimizers of the $W^{s,p}$-energy for $s\in (0,1)$ and $p\in (1,\infty )$ to those of the $W^{s,1}$-energy, both in the pointwise sense and by means of $\Gamma $-convergence. We also address the convergence of the corresponding Euler–Lagrange equations and the equivalence between minimizers and weak solutions. As ancillary results, we study some regularity issues regarding minimizers of the $W^{s,1}$-energy.

Inexact and Stochastic Generalized Conditional Gradient with Augmented Lagrangian and Proximal Step

In this paper we propose and analyze inexact and stochastic versions of the CGALP algorithm developed in [25], which we denote ICGALP , that allow for errors in the computation of several important quantities. In particular this allows one to compute some gradients, proximal terms, and/or linear minimization oracles in an inexact fashion that facilitates the practical application of the algorithm to computationally intensive settings, e.g., in high (or possibly infinite) dimensional Hilbert spaces commonly found in machine learning problems. The algorithm is able to solve composite minimization problems involving the sum of three convex proper lower-semicontinuous functions subject to an affine constraint of the form Ax = b for some bounded linear operator A. Only one of the functions in the objective is assumed to be differentiable, the other two are assumed to have an accessible proximal operator and a linear minimization oracle. As main results, we show convergence of the Lagrangian values (so-called convergence in the Bregman sense) and asymptotic feasibility of the affine constraint as well as strong convergence of the sequence of dual variables to a solution of the dual problem, in an almost sure sense. Almost sure convergence rates are given for the Lagrangian values and the feasibility gap for the ergodic primal variables. Rates in expectation are given for the Lagrangian values and the feasibility gap subsequentially in the pointwise sense. Numerical experiments verifying the predicted rates of convergence are shown as well.

Constrained estimation using penalization and MCMC

We study inference for parameters defined by either classical extremum estimators or Laplace-type estimators subject to general nonlinear constraints on the parameters. We show that running MCMC on the penalized version of the problem offers a computationally attractive alternative to solving the original constrained optimization problem. Bayesian credible intervals are asymptotically valid confidence intervals in a pointwise sense, providing exact asymptotic coverage for general functions of the parameters. We allow for nonadaptive and adaptive penalizations using the ℓp for p⩾1 penalty functions. These methods are motivated by and include as special cases model selection and shrinkage methods such as the LASSO and its Bayesian and adaptive versions. A simulation study validates the theoretical results. We also provide an empirical application on estimating the joint density of U.S. real consumption and asset returns subject to Euler equation constraints in a CRRA asset pricing model.

Pointwise error estimate in difference setting for the two-dimensional nonlinear fractional complex Ginzburg-Landau equation

In this paper, we propose a three-level linearized implicit difference scheme for the two-dimensional spatial fractional nonlinear complex Ginzburg-Landau equation. We prove that the difference scheme is stable and convergent under mild conditions. The optimal convergence order $\mathcal {O}(\tau ^{2}+{h_{x}^{2}}+{h_{y}^{2}})$ is obtained in the pointwise sense by developing a new two-dimensional fractional Sobolev imbedding inequality based on the work in Kirkpatrick et al. (Commun. Math. Phys. 317, 563–591 2013), an energy argument and careful attention to the nonlinear term. Numerical examples are presented to verify the validity of the theoretical results for different choices of the fractional orders α and β.

Asymptotic decay for defocusing semilinear wave equations in $$\mathbb {R}^{1+1}$$

This paper is devoted to the study of asymptotic behaviors of solutions to the one-dimensional defocusing semilinear wave equations. We prove that finite energy solution tends to zero in the pointwise sense, hence improving the averaged decay of Lindblad and Tao [4]. Moreover, for sufficiently localized data belonging to some weighted energy space, the solution decays in time with an inverse polynomial rate. This confirms a conjecture raised in the mentioned work. The results are based on new weighted vector fields as multipliers applied to regions bounded by light rays. The key observation for the first result is an integrated local energy decay for the potential energy, while the second result relies on a type of weighted Gagliardo-Nirenberg inequality.