Flux Norm Approach to Finite Dimensional Homogenization Approximations with Non-Separated Scales and High Contrast

We consider linear divergence-form scalar elliptic equations and vectorial equations for elasticity with rough (L ^∞(Ω), W ì \mathbb R^d{\Omega \subset \mathbb R^d}) coefficients a(x) that, in particular, model media with non-separated scales and high contrast in material properties. While the homogenization of PDEs with periodic or ergodic coefficients and well separated scales is now well understood, we consider here the most general case of arbitrary bounded coefficients. For such problems, we introduce explicit and optimal finite dimensional approximations of solutions that can be viewed as a theoretical Galerkin method with controlled error estimates, analogous to classical homogenization approximations. In particular, this approach allows one to analyze a given medium directly without introducing the mathematical concept of an e{\epsilon} family of media as in classical homogenization. We define the flux norm as the L ² norm of the potential part of the fluxes of solutions, which is equivalent to the usual H ¹-norm. We show that in the flux norm, the error associated with approximating, in a properly defined finite-dimensional space, the set of solutions of the aforementioned PDEs with rough coefficients is equal to the error associated with approximating the set of solutions of the same type of PDEs with smooth coefficients in a standard space (for example, piecewise polynomial). We refer to this property as the transfer property. A simple application of this property is the construction of finite dimensional approximation spaces with errors independent of the regularity and contrast of the coefficients and with optimal and explicit convergence rates. This transfer property also provides an alternative to the global harmonic change of coordinates for the homogenization of elliptic operators that can be extended to elasticity equations. The proofs of these homogenization results are based on a new class of elliptic inequalities. These inequalities play the same role in our approach as the div-curl lemma in classical homogenization.     

Stability, Instability, and Error of the Force-based Quasicontinuum Approximation

Due to their algorithmic simplicity and high accuracy, force-based model coupling techniques are popular tools in computational physics. For example, the force-based quasicontinuum (QCF) approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model. In this paper, we present a detailed stability and error analysis of this method. Our optimal order error estimates provide a theoretical justification for the high accuracy of the QCF approximation: they clearly demonstrate that the computational efficiency of continuum modeling can be utilized without a significant loss of accuracy if defects are captured in the atomistic region. The main challenge we need to overcome is the fact that the linearized QCF operator is typically not positive definite. Moreover, we prove that no uniform inf-sup stability condition holds for discrete versions of the W ^1,p -W ^1,q “duality pairing” with 1/p + 1/q = 1, if 1 ≤ p < ∞. However, we were able to establish an inf-sup stability condition for a discrete version of the W ^1,∞-W ^1,1 “duality pairing” which leads to optimal order error estimates in a discrete W ^1,∞-norm.     

Representation of discrete sequences with N-dimensional iterated function systems in tensor form

Iterated Function System (IFS) models have been used to represent discrete sequences where the attractor of the IFS is self-affine or piecewise self-affine in R ² or R ³ (R is the set of real numbers). In this paper, the piecewise hidden-variable fractal model is extended from R ³ to R ⁿ (n is an integer greater than 3), which is called the multi-dimensional piecewise hidden variable fractal model. This new model uses a “mapping partial derivative” and a constrained inverse algorithm to identify the model parameters. The model values depend continuously on all the hidden variables. Therefore the result is very general. Moreover, the piecewise hidden-variable fractal model in tensor form is more terse than in the usual matrix form.     

Convergence Rates in <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> for Elliptic Homogenization Problems

We study rates of convergence of solutions in L ² and H ^1/2 for a family of elliptic systems {L_e}{\{\mathcal{L}_\varepsilon\}} with rapidly oscillating coefficients in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov eigenvalues of {L_e}{\{\mathcal{L}_\varepsilon\}} . Most of our results, which rely on the recently established uniform estimates for the L ² Dirichlet and Neumann problems in Kenig and Shen (Math Ann 350:867–917, 2011; Commun Pure Appl Math 64:1–44, 2011) are new even for smooth domains.     

Temporal and Spatial Decays for the Stokes Flow

We estimate the time decay rates in L ¹, in the Hardy space and in L ^∞ of the gradient of solutions for the Stokes equations on the half spaces. For the estimates in the Hardy space we adopt the ideas in [7], and also use the heat kernel and the solution formula for the Stokes equations. We also estimate the temporal-spatial asymptotic estimates in L ^q , 1 < q < ∞, for the Stokes solutions. This work was supported by grant No. (R05-2002-000-00002-0(2002)) from the Basic Research Program of the Korea Science & Engineering Foundation.     

Slip coefficients and macroparameter jumps of a diatomic gas with rotational degrees of freedom on a weakly curved spherical surface

Using the half-space moment method, the problem of the slip of a diatomic gas along a rigid spherical surface is solved within the framework of a model kinetic equation previously proposed which takes into account the rotational degrees of freedom of the gas. Second-order slip coefficients (correctionsC _m ^′ , β _R ^′ , and β _R to the isothermal and thermal slip which are linear with respect to the Knudsen number Kn) are obtained. The gas macroparameter jump coefficientsC _v andC _q, which are of the second order in the Knudsen number and characterize the discontinuity of the normal mass and heat fluxes on the gas-rigid phase interface, are calculated. These coefficients are given as functions of the tangential momentum accommodation coefficient, the translational and rotational energy accommodation coefficients, and the Prandtl number. The coefficients are calculated for certain diatomic gases. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 163–173, January–February, 2000.     

On the completeness of oscillation spaces

The oscillation spaces introduced by Jaffard are a variation on the definition of Besov spaces for either s ≥ 0 or s ≤ −d/p. On the contrary, the spaces for −d/p < s < 0 cannot be sharply imbedded between Besov spaces with almost the same exponents, and, thus, they are new spaces of really different nature. Their norms take into account correlations between the positions of large wavelet coefficients through the scales. Several numerical studies uncovered such correlations in several settings including turbulence, image processing, traffic, finance, etc. These spaces allow one to capture oscillatory behaviors that are left undetected by Sobolev or Besov spaces. Unlike Sobolev spaces (respectively, Besov spaces B _p ^s,q (ℝ^d)), which are expressed by simple conditions on wavelet coefficients as ℓ^p norms (respectively, mixed ℓ^p − ℓ^q norms), oscillation spaces are written as ℓ^p averages of local C ^s ′ norms. In this paper, we prove the completeness of oscillation spaces in spite of such a mixture of two norms of different kinds. Published in Neliniini Kolyvannya, Vol. 8, No. 4, pp. 435–443, October–December, 2005.     

Pointwise Estimates and Stability for Dispersive–Diffusive Shock Waves

We study the stability and pointwise behavior of perturbed viscous shock waves for a general scalar conservation law with constant diffusion and dispersion. Along with the usual Lax shocks, such equations are known to admit undercompressive shocks. We unify the treatment of these two cases by introducing a new wave-tracking method based on “instantaneous projection”, giving improved estimates even in the Lax case. Another important feature connected with the introduction of dispersion is the treatment of a non-sectorial operator. An immediate consequence of our pointwise estimates is a simple spectral criterion for stability in all L ^p norms, p≥ 1 for the Lax case and p > 1 for the undercompressive case. Our approach extends immediately to the case of certain scalar equations of higher order, and would also appear suitable for extension to systems. Accepted May 29, 2000?Published online November 16, 2000     

Reynolds' analogy for the thermal convection driven by nonisothermal stretching surfaces

In the present paper the steady boundary-layer flows induced by permeable stretching surfaces with variable temperature distribution are investigated under the aspect of Reynolds' analogy r = St _x /C _f(x). It is shown that for certain stretching velocities and wall temperature distributions, “Reynolds' function”r, i.e. the ratio of the local Stanton number St _x and the skin friction coefficient C _f(x) equals −1/2 for any value of the Prandtl number Pr and of the dimensionless suction/injection velocity f _w. In all of these cases, the dimensionless temperature field ϑ is connected to the dimensionless downstream velocity f ^′ by the simple relationship ϑ=(f ^′)^Pr. It is also shown that in the general case, Reynolds' function r may possess several singularities in f _w. The largest of them represents a critical value, so that for f _w<f _w,crit the solutions of the energy equation (although they still satisfy all the boundary conditions) become nonphysical.     

A correlation for free convection heat transfer from vertical wavy surfaces

Free convection heat transfer along an isothermal vertical wavy surface was studied experimentally and numerically. A Mach-Zehnder Interferometer was used in the experiment to determine the local heat transfer coefficients. Experiments were done for three different amplitude–wavelength ratios of α = 0.05, 0.1, 0.2 and the Rayleigh numbers ranging from Ra _l = 2.9 × 10⁵ to 5.8 × 10⁵. A finite-volume based code was developed to verify the experimental study and obtain the results for all the amplitude–wavelength ratios between α = 0 to 0.2. It is found that the numerical results agree well with the experimental data. Results indicate that the frequency of the local heat transfer rate is the same as that of the wavy surface. The average heat transfer coefficient decreases as the amplitude–wavelength ratio increases and there is a significant difference between the average heat transfer coefficients of the surface with α = 0.2 and those surfaces with α = 0.05 and 0.1. The experimental data are correlated with a single equation which gives the local Nusselt number along the wavy surface as a function of the amplitude–wavelength ratio and the Rayleigh number.     

Effect of nonuniform accommodation coefficient distribution on the couette flow

The Couette flow is considered for surfaces with nonuniformly distributed energy accommodation coefficients α. It is shown that at Knudsen numbers greater or of the order of unity heat fluxes and viscous stresses can be considerably optimized by varying the surface distribution of α at a fixed integral value. At the same time, for Kn ≪ 1 the flows with nonuniformly distributed α are similar with the flow with a constant accommodation coefficient equal to its mean value.     

Existence of Compressible Current-Vortex Sheets: Variable Coefficients Linear Analysis

We study the initial-boundary value problem resulting from the linearization of the equations of ideal compressible magnetohydrodynamics and the Rankine-Hugoniot relations about an unsteady piecewise smooth solution. This solution is supposed to be a classical solution of the system of magnetohydrodynamics on either side of a surface of tangential discontinuity (current-vortex sheet). Under some assumptions on the unperturbed flow, we prove an energy a priori estimate for the linearized problem. Since the tangential discontinuity is characteristic, the functional setting is provided by the anisotropic weighted Sobolev space W₂^1,σ. Despite the fact that the constant coefficients linearized problem does not meet the uniform Kreiss-Lopatinskii condition, the estimate we obtain is without loss of smoothness even for the variable coefficients problem and nonplanar current-vortex sheets. The result of this paper is a necessary step in proving the local-in-time existence of current-vortex sheet solutions of the nonlinear equations of magnetohydrodynamics.     

Decomposition of the Deformations of a Thin Shell. Asymptotic Behavior of the Green-St Venant’s Strain Tensor

We investigate the behavior of the deformations of a thin shell, whose thickness δ tends to zero, through a decomposition technique of these deformations. The terms of the decomposition of a deformation v are estimated in terms of the L ²-norm of the distance from ∇ v to SO(3). This permits in particular to derive accurate nonlinear Korn’s inequalities for shells (or plates). Then we use this decomposition technique and estimates to give the asymptotic behavior of the Green-St Venant’s strain tensor when the “strain energy” is of order less than δ ^3/2.     

Incompressible Fluids in Thin Domains with Navier Friction Boundary Conditions (I)

This study is motivated by problems arising in oceanic dynamics. Our focus is the Navier–Stokes equations in a three-dimensional domain Ω_ɛ, whose thickness is of order O(ɛ) as ɛ → 0, having non-trivial topography. The velocity field is subject to the Navier friction boundary conditions on the bottom and top boundaries of Ω_ɛ, and to the periodicity condition on its sides. Assume that the friction coefficients are of order O(ɛ^3/4) as ɛ → 0. It is shown that if the initial data, respectively, the body force, belongs to a large set of H¹(Ω_ɛ), respectively, L²(Ω_ɛ), then the strong solution of the Navier–Stokes equations exists for all time. Our proofs rely on the study of the dependence of the Stokes operator on ɛ, and the non-linear estimate in which the contributions of the boundary integrals are non-trivial.     

Asymptotic computation of the inhomogeneity energy in a body in an external stress field

We consider a problem on an ellipsoidal inhomogeneity in an infinitely extended homogeneous isotropic elastic medium. The inhomogeneity differs from the ambient body in the elastic moduli (Poisson’s ratio ν and shear modulus μ) and in that it has intrinsic strains. We use the equivalent inclusion method to write out expressions for the Helmholtz and Gibbs free energy of the inhomogeneity as quadratic forms in the intrinsic strains and strains at infinity. The general expressions for the coefficients of these quadratic forms are written out as three rank four tensors characterizing the contribution to the energy by the plastic strain (ɛ _p ²), by the strain at infinity (ɛ ₀²), and (only for the Gibbs energy) by the cross term ɛ ⁰ ɛ _p .     

Natural Convection in a Cavity Filled with Porous Layers on the Top and Bottom Walls

Natural convection in a partially filled porous square cavity is numerically investigated using SIMPLEC method. The Brinkman-Forchheimer extended model was used to govern the flow in the porous medium region. At the porous-fluid interface, the flow boundary condition imposed is a shear stress jump, which includes both the viscous and inertial effects, together with a continuity of normal stress. The thermal boundary condition is continuity of temperature and heat flux. The results are presented with flow configurations and isotherms, local and average Nusselt number along the cold wall for different Darcy numbers from 10⁻¹ to 10⁻⁶, porosity values from 0.2 to 0.8, Rayleigh numbers from 10³ to 10⁷, and the ratio of porous layer thickness to cavity height from 0 to 0.50. The flow pattern inside the cavity is affected with these parameters and hence the local and global heat transfer. A modified Darcy–Rayleigh number is proposed for the heat convection intensity in porous/fluid filled domains. When its value is less than unit, global heat transfer keeps unchanged. The interfacial stress jump coefficients β ₁ and β ₂ were varied from  −1 to +1, and their effects on the local and average Nusselt numbers, velocity and temperature profiles in the mid-width of the cavity are investigated.     

Regularity Results for Nonlocal Equations by Approximation

We obtain C ^1,α regularity estimates for nonlocal elliptic equations that are not necessarily translation-invariant using compactness and perturbative methods and our previous regularity results for the translation-invariant case.     

Nonsimilarity solutions for non-Darcy mixed convection from horizontal surfaces in a porous medium

Non-Darcy mixed convection in a porous medium from horizontal surfaces with variable surface heat flux of the power-law distribution is analyzed. The entire mixed convection regime is divided into two regions. The first region covers the forced convection dominated regime where the dimensionless parameter ζ _f =Ra^* _x /Pe² _x is found to characterize the effect of buoyancy forces on the forced convection with K ^′ U _∞/ν characterizing the effect of inertia resistance. The second region covers the natural convection dominated regime where the dimensionless parameter ζ _n =Pe _x /Ra^*1/2 _x is found to characterize the effect of the forced flow on the natural convection, with (K ^′ U _∞/ν)Ra^*1/2 _x /Pe _x characterizing the effect of inertia resistance. To obtain the solution that covers the entire mixed convection regime the solution of the first regime is carried out for ζ _f =0, the pure forced convection limit, to ζ _f =1 and the solution of the second is carried out for ζ _n =0, the pure natural convection limit, to ζ _n =1. The two solutions meet and match at ζ _f =ζ _n =1, and R ^* _h =G ^* _h . Also a non-Darcy model was used to analyze mixed convection in a porous medium from horizontal surfaces with variable wall temperature of the power-law form. The entire mixed convection regime is divided into two regions. The first region covers the forced convection dominated regime where the dimensionless parameter ξ _f =Ra _x /Pe _x ^3/2 is found to measure the buoyancy effects on mixed convection with Da _x Pe _x /ɛ as the wall effects. The second region covers the natural convection dominated region where ξ _n =Pe _x /Ra _x ^2/3 is found to measure the force effects on mixed convection with Da _x Ra _x ^2/3/ɛ as the wall effects. Numerical results for different inertia, wall, variable surface heat flux and variable wall temperature exponents are presented. Received on 8 July 1996     

Asymptotic Stability of Riemann Solutions for a Class of Multidimensional Systems of Conservation Laws with Viscosity

We prove the asymptotic stability of two-state nonplanar Riemann solutions for a class of multidimensional hyperbolic systems of conservation laws when the initial data are perturbed and viscosity is added. The class considered here is those systems whose flux functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. In particular, we obtain the uniqueness of the self-similar L^∞ entropy solution of the two-state nonplanar Riemann problem. The asymptotic stability to which the main result refers is in the sense of the convergence as t→∞ in L_loc¹ of the space of directions ξ = x/t. That is, the solution u(t, x) of the perturbed problem satisfies u(t, tξ)→R(ξ) as t→∞, in L_loc¹(ℝⁿ), where R(ξ) is the self-similar entropy solution of the corresponding two-state nonplanar Riemann problem.