Analysis of the heterogeneous multiscale method for elliptic homogenization problems

A comprehensive analysis is presented for the heterogeneous multiscale method (HMM for short) applied to various elliptic homogenization problems. These problems can be either linear or nonlinear, with deterministic or random coefficients. In most cases considered, optimal estimates are proved for the error between the HMM solutions and the homogenized solutions. Strategies for retrieving the microstructural information from the HMM solutions are discussed and analyzed.

     

Analysis of the heterogeneous multiscale method for parabolic homogenization problems

The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution.

     

A posteriori error estimates in quantities of interest for the finite element heterogeneous multiscale method

We present an “a posteriori” error analysis in quantities of interest for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. The multiscale method is based on a macro‐to‐micro formulation, where the macroscopic physical problem is discretized in a macroscopic finite element space, and the missing macroscopic data are recovered on‐the‐fly using the solutions of corresponding microscopic problems. We propose a new framework that allows to follow the concept of the (single‐scale) dual‐weighted residual method at the macroscopic level in order to derive a posteriori error estimates in quantities of interests for multiscale problems. Local error indicators, derived in the macroscopic domain, can be used for adaptive goal‐oriented mesh refinement. These error indicators rely only on available macroscopic and microscopic solutions. We further provide a detailed analysis of the data approximation error, including the quadrature errors. Numerical experiments confirm the efficiency of the adaptive method and the effectivity of our error estimates in the quantities of interest. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A posteriori error majorants of the modeling errors for elliptic homogenization problems

In this paper, we derive new two-sided a posteriori estimates of the modeling errors for linear elliptic boundary value problems with periodic coefficients solved by homogenization. Our approach is based on the concept of functional a posteriori error estimation. The estimates are obtained for the energy norm and use solely the global flux of the non-oscillatory solution of the homogenized model and solution of a boundary value problem on the cell of periodicity.     

A posteriori error analysis for eigenvalue problems

A new a posteriori error analysis for symmetric eigenvalue problems of the adaptive finite element method (AFEM) is presented. Based on an H¹ stable L² projection, we prove reliability of the edge contribution for P₁ finite element methods. Hence the AFEM proposed can ignore the volume contribution as well as oscillations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains

In this contribution we analyze a generalization of the heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains. The method was originally introduced by E and Engquist (Commun Math Sci 1(1):87–132, 2003) for homogenization problems in fixed domains. It is based on a standard finite element approach on the macroscale, where the stiffness matrix is computed by solving local cell problems on the microscale. A-posteriori error estimates are derived in L ²(Ω) by reformulating the problem into a discrete two-scale formulation (see also, Ohlberger in Multiscale Model Simul 4(1):88–114, 2005) and using duality methods afterwards. Numerical experiments are given in order to numerically evaluate the efficiency of the error estimate.     

A posteriori error analysis for defect correction method for two parameter singular perturbation problems

Defect correction method is used for two parameter singular perturbation problem on Bakhvalov-Shishkin mesh. Use of defect correction method on Bakhvalov-Shishkin mesh gives a second order convergence. A posteriori error estimate is obtained. The numerical examples are given to establish the second order convergence in practice.     

A posteriori error analysis of a quadratic finite volume method for nonlinear elliptic problems

In this article, we construct and analyze a residual-based a posteriori error estimator for a quadratic finite volume method (FVM) for solving nonlinear elliptic partial differential equations with homogeneous Dirichlet boundary conditions. We shall prove that the a posteriori error estimator yields the global upper and local lower bounds for the norm error of the FVM. So that the a posteriori error estimator is equivalent to the true error in a certain sense. Numerical experiments are performed to illustrate the theoretical results.     

A posteriori error analysis for higher order dissipative methods for evolution problems

We prove a posteriori error estimates for time discretizations by the discontinuous Galerkin method dG(q) and the corresponding implicit Runge- Kutta-Radau method IRK-R(q) of arbitrary order q≥0 for both linear and nonlinear evolution problems of the form , with γ²-angle bounded operator . The key ingredient is a novel higher order reconstruction of the discrete solution U, which restores continuity and leads to the differential equation for a suitable interpolation operator Π and piecewise polynomial approximation F of f. We discuss applications to linear PDE, such as the convection-diffusion equation (γ ≥ 1/2) and the wave equation (formally γ = ∞), and nonlinear PDE corresponding to subgradient operators (γ = 1), such as the p-Laplacian, as well as Lipschitz operators (γ ≥ 1/2). We also derive conditional a posteriori error estimates for the time-dependent minimal surface problem.Partially supported by NSF Grants DMS-9971450 and DMS-0204670 and the General Research Board of the University of Maryland.     

The heterogeneous multiscale finite element method FE-HMM for the homogenization of linear elastic solids

The objective of the present paper is a formulation of the Heterogeneous Multiscale Finite Element Method (FE-HMM) for the homogenization of linear elastic solids in a geometrical linear frame, and doing so, for the first time, of a vector-valued field problem. The macro stiffness is estimated by stiffness sampling on heterogeneous microdomains in terms of a modified quadrature formula, which implies an equivalence of energy densities of the microscale with the macroscale. Existing a-priori estimates are assessed and used for optimal micro-macro refinement strategies in the H¹-norm and the L²-norm. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A posteriori error estimator for eigenvalue problems by mixed finite element method

In this paper,a residual type of a posteriori error estimator for the general second order elliptic eigenpair approximation by the mixed finite element method is derived and analyzed,based on a type of superconvergence result of the eigenfunction approximation.Its efficiency and reliability are proved by both theoretical analysis and numerical experiments.     

A posteriori error estimation for nonlinear variational problems

We introduce a new modus operandi for a posteriori error estimation for nonlinear (and linear) variational problems based on the duality theory of the calculus of variations. We derive what we call duality error estimates and show that they yield computable a posteriori error estimates without directly solving the dual problem.     

A posteriori error estimate for a modified weak Galerkin method solving elliptic problems

A residual‐type a posteriori error estimator is proposed and analyzed for a modified weak Galerkin finite element method solving second‐order elliptic problems. This estimator is proven to be both reliable and efficient because it provides computable upper and lower bounds on the actual error in a discrete H¹‐norm. Numerical experiments are given to illustrate the effectiveness of the this error estimator. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 381–398, 2017     

A posteriori error analysis for linearization of nonlinear elliptic problems and their discretizations

The paper is devoted to a posteriori quantitative analysis for errors caused by linearization of non-linear elliptic boundary value problems and their finite element realizations. We employ duality theory in convex analysis to derive computable bounds on the difference between the solution of a non-linear problem and the solution of the linearized problem, by using the solution of the linearized problem only. We also derive computable bounds on differences between finite element solutions of the nonlinear problem and finite element solutions of the linearized problem, by using finite element solutions of the linearized problem only. Numerical experiments show that our a posteriori error bounds are efficient.     

A posteriori error estimator competition for conforming obstacle problems

This article on the a posteriori error analysis of the obstacle problem with affine obstacles and Courant finite elements compares five classes of error estimates for accurate guaranteed error control. To treat interesting computational benchmarks, the first part extends the Braess methodology from 2005 of the resulting a posteriori error control to mixed inhomogeneous boundary conditions. The resulting guaranteed global upper bound involves some auxiliary partial differential equation and leads to four contributions with explicit constants. Their efficiency is examined affirmatively for five benchmark examples. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A posteriori error estimates for H 1-Galerkin mixed finite-element method for parabolic problems

The purpose of this article is to derive a posteriori error estimates for the H ¹-Galerkin mixed finite element method for parabolic problems. We study both semidiscrete and fully discrete a posteriori error analyses using standard energy argument. A fully discrete a posteriori error analysis based on the backward Euler method is analysed and upper bounds for the errors are derived. The estimators yield upper bounds for the errors which are global in space and time. Our analysis is based on residual approach and the estimators are free from edge residuals.     

A posteriori error bounds for the empirical interpolation method

We present rigorous a posteriori error bounds for the Empirical Interpolation Method (EIM). The essential ingredients are (i) analytical upper bounds for the parametric derivatives of the function to be approximated, (ii) the EIM “Lebesgue constant,” and (iii) information concerning the EIM approximation error at a finite set of points in parameter space. The bound is computed “off-line” and is valid over the entire parameter domain; it is thus readily employed in (say) the “on-line” reduced basis context. We present numerical results that confirm the validity of our approach.     

A type of new posteriori error estimators for stokes problems

1. IntroductionIn the numerical approximation of PDE, it is often very importals to detect regionswhere the accuracy of the numerical solution is degraded by local singularities of the solutionof the continuous problem such as the singularity near the re-entrant corller. An obviousremedy is to refine the discretization in the critical regions, i.e., to place more gridpointswhere the solution is less regular. The question is how to identify these regions automdticallyand how to determine a goo…     

A posteriori error estimation for a defect correction method applied to conduction convection problems

We present a posteriori error estimate for a defect correction method for approximating solutions of the stationary conduction convection problems in two dimension. The defect correction method is aiming at small viscosity ν. A reliable a posteriori error estimation is derived for the defect correction method. Finally, two numerical examples validate our theoretical results. The first example is a problem with known solution and the second example is a physical model of square cavity stationary flow. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

A posteriori error analysis of the discontinuous finite element methods for first order hyperbolic problems

In this paper, we consider the a posteriori error analysis of discontinuous Galerkin finite element methods for the steady and nonsteady first order hyperbolic problems with inflow boundary conditions. We establish several residual-based a posteriori error estimators which provide global upper bounds and a local lower bound on the error. Further, for nonsteady problem, we construct a fully discrete discontinuous finite element scheme and derive the a posteriori error estimators which yield global upper bound on the error in time and space. Our a posteriori error analysis is based on the mesh-dependent a priori estimates for the first order hyperbolic problems. These a posteriori error analysis results can be applied to develop the adaptive discontinuous finite element methods.