# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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## Discrete & Continuous Dynamical Systems - S

February 2015 , Volume 8 , Issue 1

Issue on numerical methods based on homogenization and two-scale convergence

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2015, 8(1): i-vi doi: 10.3934/dcdss.2015.8.1i +[Abstract](3140) +[PDF](333.4KB)
Abstract:
In this note, a classification of Homogenization-Based Numerical Methods and (in particular) of Numerical Methods that are based on the Two-Scale Convergence is done. In this classification stand: Direct Homogenization-Based Numerical Methods; H-Measure-Based Numerical Methods; Two-Scale Numerical Methods and TSAPS: Two-Scale Asymptotic Preserving Schemes.
2015, 8(1): 1-27 doi: 10.3934/dcdss.2015.8.1 +[Abstract](2489) +[PDF](734.3KB)
Abstract:
We consider a nonlinear convex stochastic homogenization problem, in a stationary setting. In practice, the deterministic homogenized energy density is approximated by a random apparent energy density, obtained by solving the corrector problem on a truncated domain.
We show that the technique of antithetic variables can be used to reduce the variance of the computed quantities, and thereby decrease the computational cost at equal accuracy. This leads to an efficient approach for approximating expectations of the apparent homogenized energy density and of related quantities.
The efficiency of the approach is numerically illustrated on several test cases. Some elements of analysis are also provided.
2015, 8(1): 29-44 doi: 10.3934/dcdss.2015.8.29 +[Abstract](2269) +[PDF](2720.1KB)
Abstract:
Averaging coefficient in a second order elliptic equation is a well known and important model problem. Additional to non-periodic rapid oscillations, the coefficient may contain barriers and channels - long and narrow bodies with low or high values of the coefficient. When the length of such structures is comparable with the problem size - there is no scale separation.
In this article we consider coefficients with barriers. We show how the averaged coefficient may be inadequate near the barriers and propose a generalization which can detect the potential problems and improve the accuracy of the averaged solution.
2015, 8(1): 45-54 doi: 10.3934/dcdss.2015.8.45 +[Abstract](2905) +[PDF](1226.5KB)
Abstract:
This paper is in the continuity of a work program, initiated in Frénod & Goubert [4], Frénod & Rousseau [5] and Bernard, Frénod & Rousseau [2]. Its goal is to develop an approach of the paralic confinement usable from the modeling slant, before implementing it in numerical tools.
More specifically, we here deal with the multiscale aspect of the confinement. If a paralic environment is separated into two (or more) connected areas, we will show that is possible to split the confinement problem into two related problems, one for each area. At the end of this paper, we will focus on the importance of the interface length between the two subdomains.
2015, 8(1): 55-76 doi: 10.3934/dcdss.2015.8.55 +[Abstract](2100) +[PDF](490.4KB)
Abstract:
In the repository, multi-physics processes are induced due to the long-time heat-emitting from the nuclear waste, which is modeled as a nonlinear system with oscillating coefficients. In this paper we first derive the homogenized system for the thermal-hydro-mass transfer processes by the technique of two-scale convergence, then present some error estimates for the first order expansions.
2015, 8(1): 77-90 doi: 10.3934/dcdss.2015.8.77 +[Abstract](1960) +[PDF](207.7KB)
Abstract:
This paper introduces a new tool so called Multi-scales H-measures to analyse the effect of heterogeneities occurring at several scales. In a first place, it recalls the course that brought the introduction of new tools for homogenization and it recalls what are H-Measures. Then the paper gives the definition and the framework of Semi-Classical Measures, presents their capability, and illustrates some of their limitations. Finally, it introduces the concept of Multi-Scale H-measures.
2015, 8(1): 91-118 doi: 10.3934/dcdss.2015.8.91 +[Abstract](3297) +[PDF](2311.7KB)
Abstract:
The reduced basis finite element heterogeneous multiscale method (RB-FE-HMM) for a class of nonlinear homogenization elliptic problems of nonmonotone type is introduced. In this approach, the solutions of the micro problems needed to estimate the macroscopic data of the homogenized problem are selected by a greedy algorithm and computed in an offline stage. It is shown that the use of reduced basis (RB) for nonlinear numerical homogenization reduces considerably the computational cost of the finite element heterogeneous multiscale method (FE-HMM). As the precomputed microscopic functions depend nonlinearly on the macroscopic solution, we introduce a new a posteriori error estimator for the greedy algorithm that guarantees the convergence of the online Newton method. A priori error estimates and uniqueness of the numerical solution are also established. Numerical experiments illustrate the efficiency of the proposed method.
2015, 8(1): 119-150 doi: 10.3934/dcdss.2015.8.119 +[Abstract](2748) +[PDF](1889.8KB)
Abstract:
In this work we introduce and analyse a new adaptive Petrov-Galerkin heterogeneous multiscale finite element method (HMM) for monotone elliptic operators with rapid oscillations. In a general heterogeneous setting we prove convergence of the HMM approximations to the solution of a macroscopic limit equation. The major new contribution of this work is an a-posteriori error estimate for the $L^2$-error between the HMM approximation and the solution of the macroscopic limit equation. The a posteriori error estimate is obtained in a general heterogeneous setting with scale separation without assuming periodicity or stochastic ergodicity. The applicability of the method and the usage of the a posteriori error estimate for adaptive local mesh refinement is demonstrated in numerical experiments. The experimental results underline the applicability of the a posteriori error estimate in non-periodic homogenization settings.
2015, 8(1): 151-168 doi: 10.3934/dcdss.2015.8.151 +[Abstract](3238) +[PDF](2811.4KB)
Abstract:
In this paper we consider the model built in [3] for short term dynamics of dunes in tidal area. We construct a Two-Scale Numerical Method based on the fact that the solution of the equation which has oscillations Two-Scale converges to the solution of a well-posed problem. This numerical method uses on Fourier series.
2015, 8(1): 169-183 doi: 10.3934/dcdss.2015.8.169 +[Abstract](3088) +[PDF](667.3KB)
Abstract:
In the framework of a Particle-In-Cell scheme for some 1D Vlasov-Poisson system depending on a small parameter, we propose a time-stepping method which is numerically uniformly accurate when the parameter goes to zero. Based on an exponential time differencing approach, the scheme is able to use large time steps with respect to the typical size of the fast oscillations of the solution.
2015, 8(1): 185-221 doi: 10.3934/dcdss.2015.8.185 +[Abstract](2227) +[PDF](740.9KB)
Abstract:
We study a Lie Transform method for a charged beam under the action of a radial external electric field. The aim of the Lie transform method that is used here is to construct a change of variable which transforms the 2D kinetic problem into a 1D problem. This reduces the dimensionality of the problem and make it easier to solve numerically. After applying the Lie transform method, we truncate the expression of the characteristics of the Vlasov equation and the expression of the Poisson equation in the Lie coordinate system and we develop a numerical method for solving the truncated model and we study its efficiency for the simulation of long time beam evolution.
2015, 8(1): 223-241 doi: 10.3934/dcdss.2015.8.223 +[Abstract](3072) +[PDF](433.2KB)
Abstract:
We develop and we explain the two-scale convergence in the covariant formalism, i.e. using differential forms on a Riemannian manifold. For that purpose, we consider two manifolds $M$ and $Y$, the first one contains the positions and the second one the oscillations. We establish some convergence results working on geodesics on a manifold. Then, we apply this framework on examples.

2020 Impact Factor: 2.425
5 Year Impact Factor: 1.490
2020 CiteScore: 3.1