ISSN:

1937-1632

eISSN:

1937-1179

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## Discrete & Continuous Dynamical Systems - S

October 2016 , Volume 9 , Issue 5

Issue on homogenization-based numerical methods

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2016, 9(5): i-ix
doi: 10.3934/dcdss.201605i

*+*[Abstract](1770)*+*[PDF](668.7KB)**Abstract:**

This note recalls what are "Homogenization-Based Numerical Methods". Then it introduces the papers of this Special Issue. In a third section it advocates for building a project in order to build "Homogenization- Based Software for Simulation of Multi-Scale Complex Systems".

2016, 9(5): 1269-1298
doi: 10.3934/dcdss.2016051

*+*[Abstract](2130)*+*[PDF](3643.2KB)**Abstract:**

In this work, we propose a mixed finite element method for solving elliptic multiscale problems based on a localized orthogonal decomposition (LOD) of Raviart--Thomas finite element spaces. It requires to solve local problems in small patches around the elements of a coarse grid. These computations can be perfectly parallelized and are cheap to perform. Using the results of these patch problems, we construct a low dimensional multiscale mixed finite element space with very high approximation properties. This space can be used for solving the original saddle point problem in an efficient way. We prove convergence of our approach, independent of structural assumptions or scale separation. Finally, we demonstrate the applicability of our method by presenting a variety of numerical experiments, including a comparison with an MsFEM approach.

2016, 9(5): 1299-1326
doi: 10.3934/dcdss.2016052

*+*[Abstract](1790)*+*[PDF](2710.9KB)**Abstract:**

In this paper we develop and analyze a Multiscale Finite Element Method (MsFEM) for problems in porous microstructures. By solving local problems throughout the domain we are able to construct a multiscale basis that can be computed in parallel and used on the coarse-grid. Since we are concerned with solving Neumann problems, the spaces of interest are conforming spaces as opposed to recent work for the Dirichlet problem in porous domains that utilizes a non-conforming framework. The periodic perforated homogenization of the problem is presented along with corrector and boundary correction estimates. These periodic estimates are then used to analyze the error in the method with respect to scale and coarse-grid size. An MsFEM error similar to the case of oscillatory coefficients is proven. A critical technical issue is the estimation of Poincaré constants in perforated domains. This issue is also addressed for a few interesting examples. Finally, numerical examples are presented to confirm our error analysis. This is done in the setting of coarse-grids not intersecting and intersecting the microstructure in the setting of isolated perforations.

2016, 9(5): 1327-1349
doi: 10.3934/dcdss.2016053

*+*[Abstract](1977)*+*[PDF](1670.7KB)**Abstract:**

In this paper, we present the Stroboscopic Averaging Method (SAM), recently introduced in [7,8,10,13], which aims at

*numerically*solving highly-oscillatory differential equations. More specifically, we first apply SAM to the Schrödinger equation on the $1$-dimensional torus and on the real line with harmonic potential, with the aim of assessing its efficiency: as compared to the well-established standard splitting schemes, the stiffer the problem is, the larger the speed-up grows (up to a factor $100$ in our tests). The geometric properties of SAM are also explored: on very long time intervals, symmetric implementations of the method show a very good preservation of the mass invariant and of the energy. In a second series of experiments on $2$-dimensional equations, we demonstrate the ability of SAM to capture qualitatively the long-time evolution of the solution (without spurring high oscillations).

2016, 9(5): 1351-1375
doi: 10.3934/dcdss.2016054

*+*[Abstract](1304)*+*[PDF](2668.5KB)**Abstract:**

In this paper, we are concerned with the study of the

*Isotropic Diffusion Source Approximation*(IDSA) [6] of radiative transfer. After having recalled well-known limits of the radiative transfer equation, we present the IDSA and adapt it to the case of the homogeneous sphere. We then show that for this example the IDSA suffers from severe numerical difficulties. We argue that these difficulties originate in the min-max switch coupling mechanism used in the IDSA. To overcome this problem we reformulate the IDSA to avoid the problematic coupling. This allows us to access the modeling error of the IDSA for the homogeneous sphere test case. The IDSA is shown to overestimate the streaming component, hence we propose a new version of the IDSA, which is numerically shown to be more accurate than the old one. Analytical results and numerical tests are provided to support the accuracy of the new proposed approximation.

2016, 9(5): 1377-1392
doi: 10.3934/dcdss.2016055

*+*[Abstract](2202)*+*[PDF](1702.1KB)**Abstract:**

A posteriori error estimates are derived in the context of two-dimensional structural elastic shape optimization under the compliance objective. It is known that the optimal shape features are microstructures that can be constructed using sequential lamination. The descriptive parameters explicitly depend on the stress. To derive error estimates the dual weighted residual approach for control problems in PDE constrained optimization is employed, involving the elastic solution and the microstructure parameters. Rigorous estimation of interpolation errors ensures robustness of the estimates while local approximations are used to obtain fully practical error indicators. Numerical results show sharply resolved interfaces between regions of full and intermediate material density.

2016, 9(5): 1393-1420
doi: 10.3934/dcdss.2016056

*+*[Abstract](2060)*+*[PDF](1748.4KB)**Abstract:**

In this contribution we address a-posteriori error estimation in $L^\infty(L^2)$ for a heterogeneous multiscale finite element approximation of time-dependent advection-diffusion problems with rapidly oscillating coefficient functions and with a large expected drift. Based on the error estimate, we derive an algorithm for an adaptive mesh refinement. The estimate and the algorithm are validated in numerical experiments, showing applicability and good results even for heterogeneous microstructures.

2016, 9(5): 1421-1445
doi: 10.3934/dcdss.2016057

*+*[Abstract](1749)*+*[PDF](560.0KB)**Abstract:**

This paper deals with a nonlinear parabolic equation for which a local solution in time exists and then blows up in a finite time. We consider the Chipot-Weissler equation: \begin{equation*} u_{t}=u_{x x} + u^{p}-|u_{x}|^{q},\ \ x\in (-1,1);\ t>0, \ \ p>1 \text{ and } 1 \leq q < \frac{2p}{p+1}. \end{equation*} We study the numerical approximation, we show that the numerical solution converges to the continuous one under some restriction on the initial data and the parameters $p$ and $q$. Moreover, we study the numerical blow up sets and we show that although the convergence of the numerical solution is guaranteed, the numerical blow up sets are sometimes different from that of the PDE.

2016, 9(5): 1447-1473
doi: 10.3934/dcdss.2016058

*+*[Abstract](1663)*+*[PDF](565.6KB)**Abstract:**

In many physical contexts, evolution convection equations may present some very large amplitude convective terms. As an example, in the context of magnetic confinement fusion, the distribution function that describes the plasma satisfies the Vlasov equation in which some terms are of the same order as $\epsilon^{-1}$, $\epsilon \ll 1$ being the characteristic gyrokinetic period of the particles around the magnetic lines. In this paper, we aim to present a model hierarchy for modeling the distribution function for any value of $\epsilon$ by using some two-scale convergence tools. Following Frénod & Sonnendrücker's recent work, we choose the framework of a singularly perturbed convection equation where the convective terms admit either a high amplitude part or a an oscillating part with high frequency $\epsilon^{-1} \gg 1$. In this abstract framework, we derive an expansion with respect to the small parameter $\epsilon$ and we recursively identify each term of this expansion. Finally, we apply this new model hierarchy to the context of a linear Vlasov equation in three physical contexts linked to the magnetic confinement fusion and the evolution of charged particle beams.

2016, 9(5): 1475-1492
doi: 10.3934/dcdss.2016059

*+*[Abstract](1698)*+*[PDF](477.1KB)**Abstract:**

The asymptotic analysis of a one-dimensional nonsimple thermoelastic problem is considered in this paper. By a detailed spectral analysis, the asymptotic expressions for eigenvalues and eigenfunctions of the considered system are developed. It is shown that the eigenfunctions form a Riesz basis on the Hilbert space and the eigenvalues asymptotically fall on two branches. One branch is along the negative horizontal axis in the complex plane and the other branch is asymptotic to a vertical line that is parallel to the imaginary axis. This gives the spectrum-determined growth condition for the $C_0-$semigroup associated to the system, and consequently, the asymptotic and the exponential stability of the solutions are deduced. The approach developed in this paper confirms the already-existing results; furthermore, it can be extended to a larger field of applications such as coupled system of rod or beam with diffusion equation. The method will be illustrated by an example of thermoelastic beam equations with Dirichlet boundary conditions.

2016, 9(5): 1493-1520
doi: 10.3934/dcdss.2016060

*+*[Abstract](1725)*+*[PDF](959.9KB)**Abstract:**

We study a Venttsel' problem in a three dimensional fractal domain for an operator in non divergence form. We prove existence, uniqueness and regularity results of the strict solution for both the fractal and prefractal problem, via a semigroup approach. In view of numerical approximations, we study the asymptotic behaviour of the solutions of the prefractal problems and we prove that the prefractal solutions converge in the Mosco-Kuwae-Shioya sense to the (limit) solution of the fractal one.

2016, 9(5): 1521-1551
doi: 10.3934/dcdss.2016061

*+*[Abstract](1485)*+*[PDF](537.4KB)**Abstract:**

In this paper we couple a long term dynamic equation of dunes of sand(LTDD) in [6] with a shallow water equation(SWE). And we study the evolution of sand dunes over long periods in the marine environment near the coast. We use works due to S. Klainerman & A. Majda [9] to show on the one hand existence and uniqueness results. On the other hand we give estimations of solutions for the dimensionless coupled system SWE-LTDD. And finally the coupled system is homogenized.

2016, 9(5): 1553-1564
doi: 10.3934/dcdss.2016062

*+*[Abstract](1699)*+*[PDF](396.9KB)**Abstract:**

In this paper we are interested in a problem of dilute emulsions of two immiscible viscous fluids, in which one is distributed in the other in the form of droplets of arbitrary shape, with non-uniform surface tension due to surfactants. The problem includes an essential kinematic condition on the droplets. In the periodic homogenization framework, it can be shown using Mosco-convergence that, as the size of the droplets converges to zero faster than the distance between the droplets, the emulsion behaves in the limit like the continuous phase. Here we determine the rate of convergence of the velocity field for the emulsion to that of the velocity for the one fluid problem and in addition, we determine the corrector in terms of the bulk and surface polarization tensors.

2016, 9(5): 1565-1574
doi: 10.3934/dcdss.2016063

*+*[Abstract](2619)*+*[PDF](391.2KB)**Abstract:**

In this work we are interested in the search of interface conditions to couple hydrostatic and nonhydrostatic ocean models. To this aim, we consider simplified systems and use a time discretization to handle linear equations. We recall the links between the two models (with the particular role of the aspect ratio $\delta=H/L\ll 1$) and introduce an iterative method based on the Schwarz algorithm (widely used in domain decomposition methods).

The convergence of this method depends strongly on the choice of interface conditions: this is why we look for exact absorbing conditions and their approximations in order to provide tractable and efficient coupling algorithms.

2016, 9(5): 1575-1590
doi: 10.3934/dcdss.2016064

*+*[Abstract](1855)*+*[PDF](412.8KB)**Abstract:**

In mathematics, homogenization theory considers the limitations of the sequences of the problems and their solutions when a parameter tends to zero. This parameter is regarded as the ratio of the characteristic size between the micro scale and macro scale. So what is considered is a sequence of problems in a fixed domain while the characteristic size in micro scale tends to zero. But in the real physics or engineering situations, the micro scale of a medium is fixed and can not be changed. In the process of homogenization, it is the size in macro scale which becomes larger and larger and tends to infinity. We observe that the homogenization in physics is not equivalent to the homogenization in mathematics up to some simple rescaling. With some direct error estimates, we explain in what sense we can accept the homogenized problem as the limitation of the original real physical problems. As a byproduct, we present some results on the mathematical homogenization of some problems with source term being only weakly compacted in $H^{-1}$, while in standard homogenization theory, the source term is assumed to be at least compacted in $H^{-1}$. A real example is also given to show the validation of our observation and results.

2018 Impact Factor: 0.545

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