SIAM Journal on Mathematical Analysis, ISSN 0036-1410, 2017, Volume 49, Issue 1, pp. 222 - 271

We develop a technique of multiple scale asymptotic expansions along mean flows and a corresponding notion of weak multiple scale convergence. These are...

Homogenization | Sigma-convergence | Strong convection regime | Two-scale convergence | Ergodic algebra with mean value | MATHEMATICS, APPLIED | sigma-convergence | HOMOGENIZATION STRUCTURES | CONVECTION | 2-SCALE CONVERGENCE | strong convection regime | GENERALIZED BESICOVITCH SPACES | DRIFT | two-scale convergence | homogenization | TRANSPORT | ergodic algebra with mean value | POROUS-MEDIA | DIFFUSION | EQUATION

Homogenization | Sigma-convergence | Strong convection regime | Two-scale convergence | Ergodic algebra with mean value | MATHEMATICS, APPLIED | sigma-convergence | HOMOGENIZATION STRUCTURES | CONVECTION | 2-SCALE CONVERGENCE | strong convection regime | GENERALIZED BESICOVITCH SPACES | DRIFT | two-scale convergence | homogenization | TRANSPORT | ergodic algebra with mean value | POROUS-MEDIA | DIFFUSION | EQUATION

Journal Article

Mathematics of Computation, ISSN 0025-5718, 2019, Volume 88, Issue 316, pp. 637 - 664

We propose a two-scale finite element method for the Monge-Ampere equation with Dirichlet boundary condition in dimension d >= 2 and prove that it converges to...

Two-scale method | Monge-Ampère equation | Monotone scheme | Regularization | Viscosity solution | Convergence | MATHEMATICS, APPLIED | Monge-Ampere equation | convergence | two-scale method | regularization | CONVEX-FUNCTIONS | viscosity solution | NUMERICAL-SOLUTION | DISCRETIZATION | MONOTONE | OPERATOR | DIRICHLET PROBLEM | monotone scheme

Two-scale method | Monge-Ampère equation | Monotone scheme | Regularization | Viscosity solution | Convergence | MATHEMATICS, APPLIED | Monge-Ampere equation | convergence | two-scale method | regularization | CONVEX-FUNCTIONS | viscosity solution | NUMERICAL-SOLUTION | DISCRETIZATION | MONOTONE | OPERATOR | DIRICHLET PROBLEM | monotone scheme

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 02/2018, Volume 41, Issue 3, pp. 1101 - 1106

We demonstrate how the problem of finding the effective property of quasiperiodic constitutive relations can be simplified to the periodic homogenization...

compactness | cut‐and‐projection | electrostatics | two‐scale convergence | quasiperiodic composites | cut-and-projection | two-scale convergence | MATHEMATICS, APPLIED | Homogenization | Constitutive relationships | Differential equations | Convergence | Photonic | Mechanics of the solides | Mechanics | Optics | Acoustics | Engineering Sciences | Physics | Naturvetenskap | Compactness | Quasiperiodic composites | Mathematics | Natural Sciences | Two-scale convergence | Matematik | Cut-and-projection | Electrostatics

compactness | cut‐and‐projection | electrostatics | two‐scale convergence | quasiperiodic composites | cut-and-projection | two-scale convergence | MATHEMATICS, APPLIED | Homogenization | Constitutive relationships | Differential equations | Convergence | Photonic | Mechanics of the solides | Mechanics | Optics | Acoustics | Engineering Sciences | Physics | Naturvetenskap | Compactness | Quasiperiodic composites | Mathematics | Natural Sciences | Two-scale convergence | Matematik | Cut-and-projection | Electrostatics

Journal Article

Applied Mathematics Letters, ISSN 0893-9659, 2010, Volume 23, Issue 10, pp. 1170 - 1173

We briefly recall the concept of multiscale convergence, which is a generalization of two-scale convergence. Then we investigate a related concept, called very...

Very weak multiscale convergence | Homogenization | Multiscale convergence | Parabolic | Two-scale convergence | PARABOLIC OPERATORS | MATHEMATICS, APPLIED | Sustainable development | Mathematics | Naturvetenskap | Natural Sciences | Matematik | Homogenization; Multiscale convergence; Parabolic; Two-scale convergence; Very weak multiscale convergence

Very weak multiscale convergence | Homogenization | Multiscale convergence | Parabolic | Two-scale convergence | PARABOLIC OPERATORS | MATHEMATICS, APPLIED | Sustainable development | Mathematics | Naturvetenskap | Natural Sciences | Matematik | Homogenization; Multiscale convergence; Parabolic; Two-scale convergence; Very weak multiscale convergence

Journal Article

Nonlinear Analysis: Real World Applications, ISSN 1468-1218, 01/2014, Volume 15, Issue 1, pp. 326 - 344

We explore the homogenization limit and rigorously derive upscaled equations for a microscopic reaction–diffusion system modeling sulfate corrosion in sewer...

Sulfate corrosion of concrete | Periodic unfolding method | Semi-linear partially dissipative system | Periodic homogenization | Two-scale convergence | Multiscale system | MATHEMATICS, APPLIED | SETS | DIFFUSION | EXCHANGE | HOMOGENIZATION | ASYMPTOTIC-BEHAVIOR | Models | Sulfates | Analysis | Corrosion and anti-corrosives | Sewer pipes | Mathematical analysis | Differential equations | Homogenization | Nonlinearity | Mathematical models | Homogenizing | Samhällsbyggnadsteknik | concrete | Matematisk analys | Civil Engineering | Teknik och teknologier | Mathematics | Mathematical Analysis | two-scale convergence | Engineering and Technology | Naturvetenskap | effective models | Natural Sciences | Matematik | sulphate corrosion | porous media

Sulfate corrosion of concrete | Periodic unfolding method | Semi-linear partially dissipative system | Periodic homogenization | Two-scale convergence | Multiscale system | MATHEMATICS, APPLIED | SETS | DIFFUSION | EXCHANGE | HOMOGENIZATION | ASYMPTOTIC-BEHAVIOR | Models | Sulfates | Analysis | Corrosion and anti-corrosives | Sewer pipes | Mathematical analysis | Differential equations | Homogenization | Nonlinearity | Mathematical models | Homogenizing | Samhällsbyggnadsteknik | concrete | Matematisk analys | Civil Engineering | Teknik och teknologier | Mathematics | Mathematical Analysis | two-scale convergence | Engineering and Technology | Naturvetenskap | effective models | Natural Sciences | Matematik | sulphate corrosion | porous media

Journal Article

Mathematical Models and Methods in Applied Sciences, ISSN 0218-2025, 05/2018, Volume 28, Issue 5, pp. 979 - 1035

The aim of this paper is to provide a complete mathematical analysis of the periodic homogenization procedure that leads to the macroscopic bidomain model in...

Bidomain model | asymptotic analysis | homogenization | two-scale convergence | MATHEMATICS, APPLIED | REACTION-DIFFUSION SYSTEMS | ELECTRIC-FIELD | SETS | EQUATIONS | CURRENTS | CELL

Bidomain model | asymptotic analysis | homogenization | two-scale convergence | MATHEMATICS, APPLIED | REACTION-DIFFUSION SYSTEMS | ELECTRIC-FIELD | SETS | EQUATIONS | CURRENTS | CELL

Journal Article

Multiscale Modeling and Simulation, ISSN 1540-3459, 2017, Volume 15, Issue 4, pp. 1651 - 1671

The aim of the paper is to introduce an alternative notion of two-scale convergence which gives a more natural modeling approach to the homogenization of...

Cell averaging | Multiscale problems | Periodic homogenization | Two-scale convergence | Boundary layers | multiscale problems | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | periodic homogenization | cell averaging | boundary layers | PHYSICS, MATHEMATICAL | two-scale convergence

Cell averaging | Multiscale problems | Periodic homogenization | Two-scale convergence | Boundary layers | multiscale problems | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | periodic homogenization | cell averaging | boundary layers | PHYSICS, MATHEMATICAL | two-scale convergence

Journal Article

International Journal for Numerical Methods in Engineering, ISSN 0029-5981, 02/2011, Volume 85, Issue 7, pp. 847 - 873

Following the theory of two‐scale convergence method introduced by Nguetseng (SIAM J. Math. Anal. 1989; 20:608–623) and further developed by Allaire (SIAM J....

Wiener polynomial chaos | Karhunen–Loève expansions | spectral methods | two‐scale convergence method | periodic homogenization | Karhunen-Loève expansions | Periodic homogenization | Two-scale convergence method | Spectral methods | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | two-scale convergence method | Karhunen-Loeve expansions | HOMOGENIZATION | Partial differential equations | Mathematical analysis | Copyrights | Gaussian | Mathematical models | Spectra | Stochasticity | Convergence

Wiener polynomial chaos | Karhunen–Loève expansions | spectral methods | two‐scale convergence method | periodic homogenization | Karhunen-Loève expansions | Periodic homogenization | Two-scale convergence method | Spectral methods | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | two-scale convergence method | Karhunen-Loeve expansions | HOMOGENIZATION | Partial differential equations | Mathematical analysis | Copyrights | Gaussian | Mathematical models | Spectra | Stochasticity | Convergence

Journal Article

ESAIM: Proceedings, ISSN 1270-900X, 12/2012, Volume 38, pp. 1 - 35

Those notes are the lecture notes of lectures given at Cemracs 2011 Summer School. I present here the classical results of Two-Scale Convergence Theory and an...

Homogenization | Partial differential equations | Lectures | Summer schools | Homogenizing | Convergence | Modeling and Simulation | Mathematics | Functional Analysis | Analysis of PDEs | Computer Science

Homogenization | Partial differential equations | Lectures | Summer schools | Homogenizing | Convergence | Modeling and Simulation | Mathematics | Functional Analysis | Analysis of PDEs | Computer Science

Journal Article

Functional Analysis and Its Applications, ISSN 0016-2663, 7/2016, Volume 50, Issue 3, pp. 204 - 218

We study the convergence of continuous spectrum eigenfunctions for differential operators of divergence type with ε-periodic coefficients, where ε is a small...

homogenization | double porosity model | Functional Analysis | Analysis | Bloch eigenfunction | Mathematics | Bloch principle | two-scale convergence | convergence of spectra | MATHEMATICS | MATHEMATICS, APPLIED | 2-SCALE CONVERGENCE | Porosity

homogenization | double porosity model | Functional Analysis | Analysis | Bloch eigenfunction | Mathematics | Bloch principle | two-scale convergence | convergence of spectra | MATHEMATICS | MATHEMATICS, APPLIED | 2-SCALE CONVERGENCE | Porosity

Journal Article

Multiscale Modeling and Simulation, ISSN 1540-3459, 2015, Volume 13, Issue 3, pp. 1061 - 1105

In this paper we generalize the periodic unfolding method and the notion of two-scale convergence on surfaces of periodic microstructures to locally periodic...

Unfolding method | Two-scale convergence | Nonperiodic microstructures | Signalling processes | Locally periodic homogenization | ERROR ESTIMATE | BEHAVIOR | unfolding method | locally periodic homogenization | PHYSICS, MATHEMATICAL | FIBER ARCHITECTURE | two-scale convergence | MECHANICS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | signalling processes | HOMOGENIZATION APPROACH | nonperiodic microstructures | BODIES | OPTIMIZATION | DOMAINS

Unfolding method | Two-scale convergence | Nonperiodic microstructures | Signalling processes | Locally periodic homogenization | ERROR ESTIMATE | BEHAVIOR | unfolding method | locally periodic homogenization | PHYSICS, MATHEMATICAL | FIBER ARCHITECTURE | two-scale convergence | MECHANICS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | signalling processes | HOMOGENIZATION APPROACH | nonperiodic microstructures | BODIES | OPTIMIZATION | DOMAINS

Journal Article

Applicable Analysis, ISSN 0003-6811, 02/2015, Volume 94, Issue 2, pp. 375 - 398

We investigate the deformation of heterogeneous plastic materials. The model uses internal variables and kinematic hardening, elastic and plastic strain are...

plasticity | homogenization | convex analysis | two-scale convergence | MATHEMATICS, APPLIED | 35B27 | 74Q10 | PRAGER MODEL | ELASTOPLASTICITY | VISCOELASTICITY | 74C05 | HYSTERESIS | MONOTONE-OPERATORS | Images | Plasticity | Mathematical models | Plastic deformation | Constraining | Homogenizing | Convergence | Strain

plasticity | homogenization | convex analysis | two-scale convergence | MATHEMATICS, APPLIED | 35B27 | 74Q10 | PRAGER MODEL | ELASTOPLASTICITY | VISCOELASTICITY | 74C05 | HYSTERESIS | MONOTONE-OPERATORS | Images | Plasticity | Mathematical models | Plastic deformation | Constraining | Homogenizing | Convergence | Strain

Journal Article

Journal of Elasticity, ISSN 0374-3535, 1/2014, Volume 114, Issue 1, pp. 69 - 84

This work focuses on performing the homogenization of the Nernst-Planck-Poisson system, in order to obtain an efficient modelling of the ionic transfer and...

Electrocapillarity | 78A57 | 35K60 | 76Sxx | Mechanics | Homogenization | Automotive Engineering | 76M50 | Nernst-Planck-Poisson system | Two-scale convergence | Physics | 35K55 | MIGRATION | BEHAVIOR | MATERIALS SCIENCE, MULTIDISCIPLINARY | ADSORPTION | TRANSPORT | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | CEMENT-BASED MATERIALS | POROUS-MEDIA | DIFFUSION | Convergence | Media | Elasticity | Models | Modelling | Homogenizing

Electrocapillarity | 78A57 | 35K60 | 76Sxx | Mechanics | Homogenization | Automotive Engineering | 76M50 | Nernst-Planck-Poisson system | Two-scale convergence | Physics | 35K55 | MIGRATION | BEHAVIOR | MATERIALS SCIENCE, MULTIDISCIPLINARY | ADSORPTION | TRANSPORT | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | CEMENT-BASED MATERIALS | POROUS-MEDIA | DIFFUSION | Convergence | Media | Elasticity | Models | Modelling | Homogenizing

Journal Article

Applicable Analysis: Homogenization and Qualitative Theory of Differential Equations dedicated to the memory of Vassily Vassilievich Zhikov, ISSN 0003-6811, 01/2019, Volume 98, Issue 1-2, pp. 64 - 90

For two-scale homogenization of a general class of asymptotically degenerating strongly elliptic symmetric PDE systems with a critically scaled high contrast...

convergence of semigroups | High-contrast homogenization | operator convergence | homogenization of high-contrast evolution problems | resolvent convergence | two-scale convergence | 35B27 | 47A10 | MATHEMATICS, APPLIED | CONVERGENCE | RESONANCES | ARTIFICIAL MAGNETISM | COMPOSITE | ELASTIC-WAVES | Operators (mathematics) | Time dependence | Boundary value problems | Electromagnetism | Functionals | Elasticity | Homogenization | Well posed problems | Convergence

convergence of semigroups | High-contrast homogenization | operator convergence | homogenization of high-contrast evolution problems | resolvent convergence | two-scale convergence | 35B27 | 47A10 | MATHEMATICS, APPLIED | CONVERGENCE | RESONANCES | ARTIFICIAL MAGNETISM | COMPOSITE | ELASTIC-WAVES | Operators (mathematics) | Time dependence | Boundary value problems | Electromagnetism | Functionals | Elasticity | Homogenization | Well posed problems | Convergence

Journal Article

Indiana University Mathematics Journal, ISSN 0022-2518, 1/2010, Volume 59, Issue 2, pp. 427 - 457

Two-scale techniques are developed for sequences of maps {uk} ⊂ Lp(Ω;ℝM) satisfying a linear differential constraint 𝒜uk = 0. These, together with...

Integers | Mathematical integrals | Homogenization | Cubes | Mathematical constants | Fourier transformations | Mathematical functions | Perceptron convergence procedure | College mathematics | Equiintegrability | Two-scale convergence | Convergence | MATHEMATICS | homogenization | REITERATED HOMOGENIZATION | CALCULUS | RELAXATION | Gamma-convergence | equiintegrability | A-QUASICONVEXITY | FUNCTIONALS | two-scale convergence | PERIODIC UNFOLDING METHOD

Integers | Mathematical integrals | Homogenization | Cubes | Mathematical constants | Fourier transformations | Mathematical functions | Perceptron convergence procedure | College mathematics | Equiintegrability | Two-scale convergence | Convergence | MATHEMATICS | homogenization | REITERATED HOMOGENIZATION | CALCULUS | RELAXATION | Gamma-convergence | equiintegrability | A-QUASICONVEXITY | FUNCTIONALS | two-scale convergence | PERIODIC UNFOLDING METHOD

Journal Article

Asymptotic Analysis, ISSN 0921-7134, 2015, Volume 91, Issue 3-4, pp. 341 - 371

In this paper we establish new homogenization results for stochastic linear hyperbolic equations with periodically oscillating coefficients. We first use the...

Prokhorov and Skorokhod compactness results | homogenization | hyperbolic stochastic PDE | corrector result | two-scale convergence | MATHEMATICS, APPLIED | 2ND-GRADE FLUIDS | 2-SCALE CONVERGENCE | CONTINUUM | WAVE-EQUATIONS | DOMAINS | ASYMPTOTIC-BEHAVIOR

Prokhorov and Skorokhod compactness results | homogenization | hyperbolic stochastic PDE | corrector result | two-scale convergence | MATHEMATICS, APPLIED | 2ND-GRADE FLUIDS | 2-SCALE CONVERGENCE | CONTINUUM | WAVE-EQUATIONS | DOMAINS | ASYMPTOTIC-BEHAVIOR

Journal Article

Applications of Mathematics, ISSN 0862-7940, 10/2018, Volume 63, Issue 5, pp. 503 - 521

This paper is devoted to the study of the linear parabolic problem $$\varepsilon\partial_t{u_\varepsilon}(x,t)-\nabla\cdot(a(x/\varepsilon,...

very weak multiscale convergence | 35K20 | 35B27 | Theoretical, Mathematical and Computational Physics | Mathematics | 2010 | multiscale convergence | Optimization | two-scale convergence | parabolic problem | homogenization | Classical and Continuum Physics | Analysis | Mathematical and Computational Engineering | Applications of Mathematics | MATHEMATICS, APPLIED | CONVERGENCE | Parabola | Convergence (Mathematics) | Homogenization | Mathematical analysis

very weak multiscale convergence | 35K20 | 35B27 | Theoretical, Mathematical and Computational Physics | Mathematics | 2010 | multiscale convergence | Optimization | two-scale convergence | parabolic problem | homogenization | Classical and Continuum Physics | Analysis | Mathematical and Computational Engineering | Applications of Mathematics | MATHEMATICS, APPLIED | CONVERGENCE | Parabola | Convergence (Mathematics) | Homogenization | Mathematical analysis

Journal Article

Multiscale Modeling and Simulation, ISSN 1540-3459, 2013, Volume 11, Issue 1, pp. 92 - 117

The introduced notion of locally periodic two-scale convergence allows one to average a wider range of microstructures, compared to the periodic one. The...

Plywood structures | Two-scale convergence | Nonperiodic microstructures | Locally periodic homogenization | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | non-periodic microstructures | BEHAVIOR | plywood structures | locally periodic homogenization | PHYSICS, MATHEMATICAL | PRINCIPLES | two-scale convergence

Plywood structures | Two-scale convergence | Nonperiodic microstructures | Locally periodic homogenization | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | non-periodic microstructures | BEHAVIOR | plywood structures | locally periodic homogenization | PHYSICS, MATHEMATICAL | PRINCIPLES | two-scale convergence

Journal Article

Japan Journal of Industral and Applied Mathematics, ISSN 0916-7005, 2012, Volume 29, Issue 2, pp. 289 - 316

We study a semi-discrete finite difference multiscale scheme for a concrete corrosion model consisting of a system of two-scale reaction–diffusion equations...

68U10 | Computational Mathematics and Numerical Analysis | Multiscale reaction–diffusion equations | Two-scale finite difference method | 35K60 | 65M06 | Approximation of weak solutions | Concrete corrosion | Mathematics | Applications of Mathematics | Convergence | 35K65 | Multiscale reaction-diffusion equations | MATHEMATICS, APPLIED | ATTACK | HOMOGENIZATION | Computer science | Models | Universities and colleges | Corrosion and anti-corrosives | Differential equations | Naturvetenskap | Computational Mathematics | Natural Sciences | Beräkningsmatematik | Matematik | finite difference method | multi scale RD systems

68U10 | Computational Mathematics and Numerical Analysis | Multiscale reaction–diffusion equations | Two-scale finite difference method | 35K60 | 65M06 | Approximation of weak solutions | Concrete corrosion | Mathematics | Applications of Mathematics | Convergence | 35K65 | Multiscale reaction-diffusion equations | MATHEMATICS, APPLIED | ATTACK | HOMOGENIZATION | Computer science | Models | Universities and colleges | Corrosion and anti-corrosives | Differential equations | Naturvetenskap | Computational Mathematics | Natural Sciences | Beräkningsmatematik | Matematik | finite difference method | multi scale RD systems

Journal Article

Asymptotic Analysis, ISSN 0921-7134, 2012, Volume 80, Issue 3-4, pp. 237 - 267

We establish a rate of convergence of the two scale expansion (in the sense of homogenization theory) of the solution to a highly oscillatory elliptic partial...

weakly stochastic homogenization | two-scale expansion | PERIODIC HOMOGENIZATION | COEFFICIENTS | MATHEMATICS, APPLIED | ERROR ESTIMATE

weakly stochastic homogenization | two-scale expansion | PERIODIC HOMOGENIZATION | COEFFICIENTS | MATHEMATICS, APPLIED | ERROR ESTIMATE

Journal Article

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