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 refined convergence result in homogenization of second order parabolic systems

We derive the sharp $O(\epsilon )$ convergence rate in $L^2(0,T;L^{q_0}(\mathrm{\Omega })),q_0=2d/(d?1)$ in periodic homogenization of second order parabolic systems with bounded measurable coefficients in Lipschitz cylinders. This extends the corresponding result for elliptic systems established in [20] to parabolic systems and improves the corresponding result in $L^2$ settings derived in [7], [28] for second order parabolic systems with time-dependent coefficients.     

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     

Multiscale stochastic homogenization of convection-diffusion equations

Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic behaviour of a sequence of realizations of the form ∂u _ɛ ^ω / ∂t+1 / ɛ ₃ C(T ₃(x/ɛ ₃)ω ₃) · ∇u _ɛ ^ω − div(α(T ₂(x/ɛ ₂)ω ₂, t) ∇u _ɛ ^ω ) = f. It is shown, under certain structure assumptions on the random vector field C(ω ₃) and the random map α(ω ₁, ω ₂, t), that the sequence {u _ɛ ^ω } of solutions converges in the sense of G-convergence of parabolic operators to the solution u of the homogenized problem ∂u/∂t − div (B(t)∇u= f).     

Homogenization of a nonlinear degenerate parabolic differential equation

This paper is devoted to the homogenization of a nonlinear degenerate parabolic problem ɑtu∈-div(D(x/∈, u∈,▽u∈)+ K(x/∈, u∈))= f(x) with Dirichlet boundary condition. Here the operator D(y, s,s) is periodic in y and degenerated in ▽s. In the paper, under the two-scale convergence theory, we obtain the limit equation as ∈→ 0 and also prove the corrector results of ▽u∈ to strong convergence.     

Reiterated homogenization of monotone parabolic problems

Reiterated homogenization is studied for divergence structure parabolic problems of the form . It is shown that under standard assumptions on the function a(y ₁,y ₂,t,ξ) the sequence of solutions converges weakly in to the solution u of the homogenized problem .      

On homogenization of periodic parabolic systems

We study homogenization in the small period limit for a periodic parabolic Cauchy problem in ^d and prove that the solutions converge in L ₂(^d) to the solution of the homogenized problem for each t > 0. For the L₂(^d)-norm of the difference, we obtain an order-sharp estimate uniform with respect to the L ₂(^d)-norm of the initial value.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 86–90, 2004Original Russian Text Copyright © by T. A. SuslinaSupported by RFBR grant No. 02-01-00798.     

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.

     

On the homogenization of degenerate parabolic equations

1. Introduction and Main ResultsLet T > 0 and fi C R" be an open bounded domain with Lipschitz boundary. Suppesthat r C on is measurable whose Hausdorff measure H"--'(r) > 0. The outward normito fi is denoted by u = (ul,' ) W). Consider the mixed boundary value problem of thfollowing degenerate parabolic equations of second order:For each s > 0, problem (I.) may be used to describe nonsteady filtration (see [1] and the references therein). The existence, uniquenss and regularity result…     

Korn's inequality for periodic solids and convergence rate of homogenization

In a three-dimensional solid with arbitrary periodic Lipschitz perforation the Korn inequality is proved with a constant independent of the perforation size. The convergence rate of homogenization as a function of the Sobolev–Slobodetskii smoothness of data is also estimated. We improve foregoing results in elasticity dropping customary restrictions on the shape of the periodicity cell and superfluous smoothness and smallness assumptions on the external forces and traction.     

Homogenization of some parabolic operators with several time scales

The main focus in this paper is on homogenization of the parabolic problem ∂ _t uɛ − ∇ · (a(x/ɛ,t/ɛ,t/ɛ ^r )∇u ^ɛ ) = f. Under certain assumptions on a, there exists a G-limit b, which we characterize by means of multiscale techniques for r > 0, r ≠ 1. Also, an interpretation of asymptotic expansions in the context of two-scale convergence is made.     

Multiscale problems and homogenization for second-order Hamilton–Jacobi equations

We prove a general convergence result for singular perturbations with an arbitrary number of scales of fully nonlinear degenerate parabolic PDEs. As a special case we cover the iterated homogenization for such equations with oscillating initial data. Explicit examples, among others, are the two-scale homogenization of quasilinear equations driven by a general hypoelliptic operator and the n-scale homogenization of uniformly parabolic fully nonlinear PDEs.     

Monotone iterates with quadratic convergence rate for solving weighted average approximations to semilinear parabolic problems

This paper deals with discrete monotone iterative methods for solving semilinear singularly perturbed parabolic problems. Monotone sequences, based on the accelerated monotone iterative method, are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem. This monotone convergence leads to the existence-uniqueness theorem. An analysis of convergence of the monotone iterative method to the solutions of the nonlinear difference scheme is given. Numerical experiments are presented.     

Anisotropic parabolic problems with Orlicz data

In this work, we prove the existence of weak solution for a class of anisotropic parabolic problems with Orlicz data. Our approach is based on the anisotropic Sobolev inequality, a smoothness, and compactness results. Copyright © 2011 John Wiley & Sons, Ltd.     

IMEX method convergence for a parabolic equation

Although implicit-explicit (IMEX) methods for approximating solutions to semilinear parabolic equations are relatively standard, most recent works examine the case of a fully discretized model. We show that by discretizing time only, one can obtain an elementary convergence result for an implicit-explicit method. This convergence result is strong enough to imply existence and uniqueness of solutions to a class of semilinear parabolic equations.     

Operator estimates in nonlinear problems of reiterated homogenization

For a nonlinear elliptic equation of monotone type with multiscale coefficients, we obtain operator-type estimates for the difference between the solution and special approximations.     

A second-order Magnus-type integrator for quasi-linear parabolic problems

In this paper, we consider an explicit exponential method of classical order two for the time discretisation of quasi-linear parabolic problems. The numerical scheme is based on a Magnus integrator and requires the evaluation of two exponentials per step. Our convergence analysis includes parabolic partial differential equations under a Dirichlet boundary condition and provides error estimates in Sobolev spaces. In an abstract formulation the initial boundary value problem is written as an initial value problem on a Banach space 

    given

involving the sectorial operator with domain independent of . Under reasonable regularity requirements on the problem, we prove the stability of the numerical method and derive error estimates in the norm of certain intermediate spaces between  and . Various applications and a numerical experiment illustrate the theoretical results.

     

Individual homogenization of nonlinear parabolic operators

In this article, we prove an individual homogenization result for a class of almost periodic nonlinear parabolic operators. The spatial and temporal heterogeneities are almost periodic functions in the sense of Besicovitch. The latter allows discontinuities and is suitable for many applications. First, we derive stability and comparison estimates for sequences of G-convergent nonlinear parabolic operators. Furthermore, using these estimates, the individual homogenization result is shown.     

Pointwise convergence of the wavelet solution to the parabolic equation with variable coefficients

We consider the parabolic equation with variable coefficients k(x)uxx = ut,0,x ≤ 1,t ≥ 0, where 0 < α ≤ k(x) < +∞, the solution on the boundary x = 0 is a given function g and ux(0,t) = 0. We use wavelet Galerkin method with Meyer multi-resolution analysis to obtain a wavelet approximating solution, and also get an estimate between the exact solution and the wavelet approximating solution of the problem.