Multiscale finite element for problems with highly oscillatory coefficients

Summary. In this paper, we study a multiscale finite element method for solving a class of elliptic problems with finite number of well separated scales. The method is designed to efficiently capture the large scale behavior of the solution without resolving all small scale features. This is accomplished by constructing the multiscale finite element base functions that are adaptive to the local property of the differential operator. The construction of the base functions is fully decoupled from element to element; thus the method is perfectly parallel and is naturally adapted to massively parallel computers. We present the convergence analysis of the method along with the results of our numerical experiments. Some generalizations of the multiscale finite element method are also discussed. Received April 17, 1998 / Revised version received March 25, 2000 / Published online June 7, 2001     

A mixed multiscale finite element method for elliptic problems with oscillating coefficients

The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large-scale structure of the solutions without resolving all the fine-scale structures. Motivated by the numerical simulation of flow transport in highly heterogeneous porous media, we propose a mixed multiscale finite element method with an over-sampling technique for solving second order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carried out for flow transport in a porous medium with a random log-normal relative permeability to demonstrate the efficiency and accuracy of the proposed method.     

Multiscale methods for elliptic homogenization problems

In this article we study two families of multiscale methods for numerically solving elliptic homogenization problems. The recently developed multiscale finite element method [Hou and Wu, J Comp Phys 134 (1997), 169–189] captures the effect of microscales on macroscales through modification of finite element basis functions. Here we reformulate this method that captures the same effect through modification of bilinear forms in the finite element formulation. This new formulation is a general approach that can handle a large variety of differential problems and numerical methods. It can be easily extended to nonlinear problems and mixed finite element methods, for example. The latter extension is carried out in this article. The recently introduced heterogeneous multiscale method [Engquist and Engquist, Comm Math Sci 1 (2003), 87–132] is designed for efficient numerical solution of problems with multiscales and multiphysics. In the second part of this article, we study this method in mixed form (we call it the mixed heterogeneous multiscale method). We present a detailed analysis for stability and convergence of this new method. Estimates are obtained for the error between the homogenized and numerical multiscale solutions. Strategies for retrieving the microstructural information from the numerical solution are provided and analyzed. Relationship between the multiscale finite element and heterogeneous multiscale methods is discussed. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A finite element method for elliptic problems with rapidly oscillating coefficients

In this paper, we consider solving second-order elliptic problems with rapidly oscillating coefficients. Under the assumption that the oscillating coefficients are periodic, on the basis of classical homogenization theory, we present a finite element method whose key is to combine a numerical approximation of the 1-order approximate solution of those equations and a numerical approximation of the classical boundary corrector of those equations from different meshes exploiting the need for different levels of resolution. Numerical experiments are included to illustrate the competitive behavior of the proposed finite element method.     

Méthode des éléments finis multi-échelles pour le problème de Stokes

This work consists of a numerical study of a multi-scale finite element method for a Stokes-type problem with highly oscillating coefficients. The objective of this method is to capture the multi-scale structure of the solution via local basis functions calculated in advance, which contain the essential information on small scales.     

矩阵形式二次修正Maxwell-Dirac系统的多尺度算法

带二次修正项的Dirac方程在拓扑绝缘体、石墨烯、超导等新材料电磁光特性分析中有着十分广泛的应用.本文工作的创新点有：一是首次提出了矩阵形式带有二次修正项的Dirac方程,它是比较一般的数学框架,涵盖了上述材料体系很多重要的物理模型,具体见附录A;二是针对上述材料体系的电磁响应问题,提出了有界区域Weyl规范下具有周期间断系数矩阵形式带二次修正项Maxwell-Dirac系统的多尺度渐近方法,结合Crank-Nicolson有限差分方法和自适应棱单元方法,发展了一类多尺度算法.数值试验结果验证了多尺度渐近方法的正确性和算法的有效性.     

Resolvent Krylov subspace approximation to operator functions

We consider the approximation of operator functions in resolvent Krylov subspaces. Besides many other applications, such approximations are currently of high interest for the approximation of φ-functions that arise in the numerical solution of evolution equations by exponential integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behaviour if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we analyse a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators whose field of values lies in the left half plane. In contrast to standard Krylov methods, the convergence will be independent of the norm of the discretised operator and thus of the spatial discretisation. We will discuss efficient implementations for finite element discretisations and illustrate our analysis with numerical experiments.     

Extended multiscale FEM for design of beams and frames with complex topology

A numerical method for design of beams and frames with complex topology is proposed. The method is based on extended multi-scale finite element method where beam finite elements are used on coarse scale and continuum elements on fine scale. A procedure for calculation of multi-scale base functions, up-scaling and downscaling techniques is proposed by using a modified version of window method that is used in computational homogenization. Coarse scale finite element is embedded into a frame of a material that is representing surrounding structure in a sense of mechanical properties. Results show that this method can capture displacements, shear deformations and local stress-strain gradients with significantly reduced computational time and memory comparing to full scale continuum model. Moreover, this method includes a special hybrid finite elements for precise modelling of structural joints. Hence, the proposed method has a potential application in large scale 2D and 3D structural analysis of non-standard beams and frames where spatial interaction between structural elements is important.     

FE heterogeneous multiscale method for long-time wave propagation

A new finite element heterogeneous multiscale method (FE-HMM) is proposed for the numerical solution of the wave equation over long times in a rapidly varying medium. Our FE-HMM captures long-time dispersive effects of the true solution at a cost similar to that of a standard numerical homogenization scheme which, however, only captures the short-time macroscale behavior of the wave field.     

Rational approximation to trigonometric operators

We consider the approximation of trigonometric operator functions that arise in the numerical solution of wave equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behavior if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we propose and analyze a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators. In contrast to standard Krylov methods, the convergence will be independent of the norm of the operator and thus of its spatial discretization. We will discuss efficient implementations for finite element discretizations and illustrate our analysis with numerical experiments. AMS subject classification (2000)  65F10, 65L60, 65M60, 65N22     

A new two grid variational multiscale method for steady‐state natural convection problem

A two‐grid variational multiscale method based on two local Gauss integrations for solving the stationary natural convection problem is presented in this article. A significant feature of the method is that we solve the natural convection problem on a coarse mesh using finite element variational multiscale method based on two local Gauss integrations firstly, and then find a fine grid solution by solving a linearized problem on a fine grid. In the computation, we introduce two local Gauss integrations as a stabilizing term to replace the projection operator without adding other variables. The stability estimates and convergence analysis of the new method are derived. Ample numerical experiments are performed to validate the theoretical predictions and demonstrate the efficiency of the new method. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiscale discontinuous Petrov–Galerkin method for the multiscale elliptic problems

In this article, we present a new multiscale discontinuous Petrov–Galerkin method (MsDPGM) for multiscale elliptic problems. This method utilizes the classical oversampling multiscale basis in the framework of a Petrov–Galerkin version of the discontinuous Galerkin method, allowing us to better cope with multiscale features in the solution. MsDPGM takes advantage of the multiscale Petrov–Galerkin method (MsPGM) and the discontinuous Galerkin method (DGM). It can eliminate the resonance error completely and decrease the computational costs of assembling the stiffness matrix, thus, allowing for more efficient solution algorithms. On the basis of a new H² norm error estimate between the multiscale solution and the homogenized solution with the first‐order corrector, we give a detailed convergence analysis of the MsDPGM under the assumption of periodic oscillating coefficients. We also investigate a multiscale discontinuous Galerkin method (MsDGM) whose bilinear form is the same as that of the DGM but the approximation space is constructed from the classical oversampling multiscale basis functions. This method has not been analyzed theoretically or numerically in the literature yet. Numerical experiments are carried out on the multiscale elliptic problems with periodic and randomly generated log‐normal coefficients. Their results demonstrate the efficiency of the proposed method.     

A two‐level variational multiscale method for incompressible flows based on two local Gauss integrations

In this article, a two‐level variational multiscale method for incompressible flows based on two local Gauss integrations is presented. We solve the Navier–Stokes problem on a coarse mesh using finite element variational multiscale method based on two local Gauss integrations, then seek a fine grid solution by solving a linearized problem on a fine grid. In computation, we use the two local Gauss integrations to replace the projection operator without adding any variables. Stability analysis is performed, and error estimates of the method are derived. Finally, a series of numerical experiments are also given, which confirm the theoretical analysis and demonstrate the efficiency of the new method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Optimal Control for Multiscale Elliptic Equations with Rough Coefficients

This paper concerns the convex optimal control problem governed by multiscale elliptic equations with arbitrarily rough $L^\infty$ coefficients, which has not only complex coupling between nonseparable scales and nonlinearity, but also important applications in composite materials and geophysics. We use one of the recently developed numerical homogenization techniques, the so-called Rough Polyharmonic Splines (RPS) and its generalization (GRPS) for the efficient resolution of the elliptic operator on the coarse scale. Those methods have optimal convergence rate which do not rely on the regularity of the coefficients nor the concepts of scale-separation or periodicity. As the iterative solution of the nonlinearly coupled OCP-OPT formulation for the optimal control problem requires solving the corresponding (state and co-state) multiscale elliptic equations many times with different right hand sides, numerical homogenization approach only requires one-time pre-computation on the fine scale and the following iterations can be done with computational cost proportional to coarse degrees of freedom. Numerical experiments are presented to validate the theoretical analysis.     

A parallel finite element variational multiscale method based on fully overlapping domain decomposition for incompressible flows

Based on fully overlapping domain decomposition and a recent variational multiscale method, a parallel finite element variational multiscale method for convection dominated incompressible flows is proposed and analyzed. In this method, each processor computes a local finite element solution in its own subdomain using a global mesh that is locally refined around its own subdomain, where a stabilization term based on two local Gauss integrations is adopted to stabilize the numerical form of the Navier–Stokes equations. Using the technical tool of local a priori estimate for the finite element solution, error bounds of the discrete solution are estimated. Algorithmic parameter scalings are derived. Numerical tests are also given to verify the theoretical predictions and demonstrate the effectiveness of the method. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 856–875, 2015     

A priori estimates for two multiscale finite element methods using multiple global fields to wave equations

We consider a scalar wave equation with nonseparable spatial scales. If the solution of the wave equation smoothly depends on some global fields, then we can utilize the global fields to construct multiscale finite element basis functions. We present two finite element approaches using the global fields: partition of unity method and mixed multiscale finite element method. We derive a priori error estimates for the two approaches and theoretically investigate the relation between the smoothness of the global fields and convergence rates of the approximations for the wave equation. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

QTT-finite-element approximation for multiscale problems I: model problems in one dimension

Tensor-compressed numerical solution of elliptic multiscale-diffusion and high frequency scattering problems is considered. For either problem class, solutions exhibit multiple length scales governed by the corresponding scale parameter: the scale of oscillations of the diffusion coefficient or smallest wavelength, respectively. As is well-known, this imposes a scale-resolution requirement on the number of degrees of freedom required to accurately represent the solutions in standard finite-element (FE) discretizations. Low-order FE methods are by now generally perceived unsuitable for high-frequency coefficients in diffusion problems and high wavenumbers in scattering problems. Accordingly, special techniques have been proposed instead (such as numerical homogenization, heterogeneous multiscale method, oversampling, etc.) which require, in some form, a-priori information on the microstructure of the solution. We analyze the approximation properties of tensor-formatted, conforming first-order FE methods for scale resolution in multiscale problems without a-priori information. The FE methods are based on the dynamic extraction of principal components from stiffness matrices, load and solution vectors by the quantized tensor train (QTT) decomposition. For prototypical model problems, we prove that this approach, by means of the QTT reparametrization of the FE space, allows to identify effective degrees of freedom to replace the degrees of freedom of a uniform “virtual” (i.e. never directly accessed) mesh, whose number may be prohibitively large to realize computationally. Precisely, solutions of model elliptic homogenization and high-frequency acoustic scattering problems are proved to admit QTT-structured approximations whose number of effective degrees of freedom required to reach a prescribed approximation error scales polylogarithmically with respect to the reciprocal of the target Sobolev-norm accuracy ε with only a mild dependence on the scale parameter. No a-priori information on the nature of the problems and intrinsic length scales of the solution is required in the numerical realization of the presently proposed QTT-structured approach. Although only univariate model multiscale problems are analyzed in the present paper, QTT structured algorithms are applicable also in several variables. Detailed numerical experiments confirm the theoretical bounds. As a corollary of our analysis, we prove that for the mentioned model problems, the Kolmogorov n-widths of solution sets are exponentially small for analytic data, independently of the problems’ scale parameters. That implies, in particular, the exponential convergence of reduced basis techniques which is scale-robust, i.e., independent of the scale parameter in the problem.     

Multiscale finite element discretizations based on local defect correction for the biharmonic eigenvalue problem of plate buckling

In this paper, we study the multiscale finite element discretizations about the biharmonic eigenvalue problem of plate buckling. On the basis of the work of Dai and Zhou (SIAM J. Numer. Anal. 46[1] [2008] 295‐324), we establish a three‐scale scheme, a multiscale discretization scheme, and the associated parallel version based on local defect correction. We first prove a local priori error estimate of finite element approximations, then give the error estimates of multiscale discretization schemes. Theoretical analysis and numerical experiments indicate that our schemes are suitable and efficient for eigenfunctions with local low smoothness.     

Multiscale modeling and analysis for some special additive noises driven stochastic partial differential equations

This paper is concerned with some special additive noises driven stochastic partial differential equations with multiscale parameters. Then, the constraint energy minimizing generalized multiscale finite element method with a novel multiscale spectral representation of the noise is constructed to solve the multiscale models. The corresponding convergence analysis and error estimates are derived, and the effects of noises on the accuracy of the multiscale computation are demonstrated. Some numerical examples are provided to validate our theoretic analysis, and numerical results show the highly efficient computational performance of our method, which is a beneficial attempt to deal with the noises in the complex multiscale stochastic physical system.     

On the finite volume element method

Summary The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH ¹ semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h ²). Results on the effects of numerical integration are also included.This research was sponsored in part by the Air Force Office of Scientific Research under grant number AFOSR-86-0126 and the National Science Foundation under grant number DMS-8704169. This work was performed while the author was at the University of Colorado at Denver