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Publication# A parabolic local problem with exponential decay of the resonance error for numerical homogenization

Résumé

This paper aims at an accurate and efficient computation of effective quantities, e.g. the homogenized coefficients for approximating the solutions to partial differential equations with oscillatory coefficients. Typical multiscale methods are based on a micro-macro-coupling, where the macromodel describes the coarse scale behavior, and the micromodel is solved only locally to upscale the effective quantities, which are missing in the macromodel. The fact that the microproblems are solved over small domains within the entire macroscopic domain, implies imposing artificial boundary conditions on the boundary of the microscopic domains. A naive treatment of these artificial boundary conditions leads to a first-order error in epsilon/delta, where epsilon < delta represents the characteristic length of the small scale oscillations and delta(d) is the size of microdomain. This error dominates all other errors originating from the discretization of the macro and the microproblems, and its reduction is a main issue in today's engineering multiscale computations. The objective of this work is to analyze a parabolic approach, first announced in A. Abdulle, D. Arjmand, E. Paganoni, C. R. Acad. Sci. Paris, Ser. I, 2019, for computing the homogenized coefficients with arbitrarily high convergence rates in epsilon/delta. The analysis covers the setting of periodic microstructure, and numerical simulations are provided to verify the theoretical findings for more general settings, e.g. non-periodic microstructures.

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Mathematical and numerical aspects of free surface flows are investigated. On one hand, the mathematical analysis of some free surface flows is considered. A model problem in one space dimension is first investigated. The Burgers equation with diffusion has to be solved on a space interval with one free extremity. This extremity is unknown and moves in time. An ordinary differential equation for the position of the free extremity of the interval is added in order to close the mathematical problem. Local existence in time and uniqueness results are proved for the problem with given domain, then for the free surface problem. A priori and a posteriori error estimates are obtained for the semi-discretization in space. The stability and the convergence of an Eulerian time splitting scheme are investigated. The same methodology is then used to study free surface flows in two space dimensions. The incompressible unsteady Navier-Stokes equations with Neumann boundary conditions on the whole boundary are considered. The whole boundary is assumed to be the free surface. An additional equation is used to describe the moving domain. Local existence in time and uniqueness results are obtained. On the other hand, a model for free surface flows in two and three space dimensions is investigated. The liquid is assumed to be surrounded by a compressible gas. The incompressible unsteady Navier-Stokes equations are assumed to hold in the liquid region. A volume-of-fluid method is used to describe the motion of the liquid domain. The velocity in the gas is disregarded and the pressure is computed by the ideal gas law in each gas bubble trapped by the liquid. A numbering algorithm is presented to recognize the bubbles of gas. Gas pressure is applied as a normal force on the liquid-gas interface. Surface tension effects are also taken into account for the simulation of bubbles or droplets flows. A method for the computation of the curvature is presented. Convergence and accuracy of the approximation of the curvature are discussed. A time splitting scheme is used to decouple the various physical phenomena. Numerical simulations are made in the frame of mould filling to show that the influence of gas on the free surface cannot be neglected. Curvature-driven flows are also considered.

Mathematical and numerical aspects of viscoelastic flows are investigated here. Two simplified mathematical models are considered. They are motivated by a splitting algorithm for solving viscoelastic flows with free surfaces. The first model is a simplified Oldroyd-B model. Existence on a fixed time interval is proved in several Banach spaces provided the data are small enough. Short time existence is also proved for arbitrarily large data in Hölder spaces for the time variable. These results are based on the maximal regularity property of the Stokes operator and on the analycity behavior of the corresponding semi-group. A finite element discretization in space is then proposed. Existence of the numerical solution is proved for small data, as well as a priori error estimates, using an implicit function theorem framework. Then, the extension of these results to a stochastic simplified Hookean dumbbells model is discussed. Because of the presence of the Brownian motion, existence in a fixed time interval, provided the data are small enough, is proved only in some of the Banach spaces considered previously. The dumbbells' elongation is split in two parts, one satisfying a standart stochastic differential equation, the other satisfying a partial differential equation with a stochastic source term. A finite element discretization in space is also proposed. Existence of the numerical solution is proved for small data, as well as a priori error estimates. A numerical algorithm for solving viscoelastic flows with free surfaces is also described. This algorithm is based on a splitting method in time and two different meshes are used for the space discretization. Convergence of the numerical model is checked for the pure extensional flow and the filling of a pipe. Then, numerical results are reported for the stretching of a filament and for jet buckling.

Doghonay Arjmand, Edoardo Paganoni

This paper aims at an accurate and ecient computation of eective quantities, e.g., the homogenized coecients for approximating the solu- tions to partial dierential equations with oscillatory coecients. Typical multiscale methods are based on a micro-macro coupling, where the macro model describes the coarse scale behaviour, and the micro model is solved only locally to upscale the eective quantities, which are missing in the macro model. The fact that the micro problems are solved over small domains within the entire macroscopic domain, implies imposing arti- cial boundary conditions on the boundary of the microscopic domains. A naive treatment of these articial boundary conditions leads to a rst order error in "=, where " < represents the characteristic length of the small scale oscillations and d is the size of micro domain. This er- ror dominates all other errors originating from the discretization of the macro and the micro problems, and its reduction is a main issue in to- day's engineering multiscale computations. The objective of the present work is to analyse a parabolic approach, rst announced in [A. Abdulle, D. Arjmand, E. Paganoni, C. R. Acad. Sci. Paris, Ser. I, 2019], for computing the homogenized coecients with arbitrarily high convergence rates in "=. The analysis covers the setting of periodic microstructure, and numerical simulations are provided to verify the theoretical ndings for more general settings, e.g. random stationary micro structures.