Localized orthogonal decomposition method for the wave equation with a continuum of scales

This paper is devoted to numerical approximations for the wave equation with a multiscale character. Our approach is formulated in the framework of the Localized Orthogonal Decomposition (LOD) interpreted as a numerical homogenization with an $L^2$-projection. We derive explicit convergence rates of the method in the $L^{\infty}(L^2)$-, $W^{1,\infty}(L^2)$- and $L^{\infty}(H^1)$-norms without any assumptions on higher order space regularity or scale-separation. The order of the convergence rates depends on further graded assumptions on the initial data. We also prove the convergence of the method in the framework of G-convergence without any structural assumptions on the initial data, i.e. without assuming that it is well-prepared. This rigorously justifies the method. Finally, the performance of the method is demonstrated in numerical experiments

Upscaling for the Laplace problem using a discontinuous Galerkin method

International audienceScientists and engineers generally tackle problems that include multiscale effects and that are thus difficult to solve numerically. The main difficulty is to capture both the fine and the coarse scales to get an accurate numerical solution. Indeed, the computations are generally performed by using numerical schemes based on grids. But the stability and thus the accuracy of the numerical method depends on the size of the grid which must be refined drastically in the case of very fine scales. That implies huge computational costs and in particular the limitations of the memory capacity are often reached. It is thus necessary to use numerical methods that are able to capture the fine scale effects with computations on coarse meshes. Operator-based upscaling is one of them and we present a first attempt to adapt that technique to a Discontinuous Galerkin Method (DGM). We consider the Laplace problem as a benchmark and we compare the performance of the resulting numerical scheme with the classical one using Lagrange finite elements. The comparison involves both an accuracy analysis and a complexity calculus. This work shows that there is an interest of combining DGM with upscaling

Homogenization of linear hyperbolic stochastic partial differential equation with rapidly oscillating coefficients : the two scale convergence method

In this paper we establish new homogenization results for stochastic linear hyperbolic equations with periodically oscillating coefficients. We first use the multiple expansion method to drive the homogenized problem. Next we use the two scale convergence method and Prokhorov’s and Skorokhod’s probabilistic compactness results. We prove that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized stochastic hyperbolic problem with constant coefficients. We also prove a corrector result.National Research Foundation of South Africa and the University of Pretoria.http://www.iospress.nlhb2016Mathematics and Applied Mathematic

Finite element heterogeneous multiscale methods for the wave equation

Wave phenomena appear in a wide range of applications such as full-waveform seismic inversion, medical imaging, or composite materials. Often, they are modeled by the acoustic wave equation. It can be solved by standard numerical methods such as, e.g., the finite element (FE) or the finite difference method. However, if the wave propagation speed varies on a microscopic length scale denoted by epsilon, the computational cost becomes infeasible, since the medium must be resolved down to its finest scale. In this thesis we propose multiscale numerical methods which approximate the overall macroscopic behavior of the wave propagation with a substantially lower computational effort. We follow the design principles of the heterogeneous multiscale method (HMM), introduced in 2003 by E and Engquist. This method relies on a coarse discretization of an a priori unknown effective equation. The missing data, usually the parameters of the effective equation, are estimated on demand by solving microscale problems on small sampling domains. Hence, no precomputation of these effective parameters is needed. We choose FE methods to solve both the macroscopic and the microscopic problems. For limited time the overall behavior of the wave is well described by the homogenized wave equation. We prove that the FE-HMM method converges to the solution of the homogenized wave equation. With increasing time, however, the true solution deviates from the classical homogenization limit, as a large secondary wave train develops. Neither the homogenized solution, nor the FE-HMM capture these dispersive effects. To capture them we need to modify the FE-HMM. Inspired by higher order homogenization techniques we additionally compute a correction term of order epsilon^2. Since its computation also relies on the solution of the same microscale problems as the original FE-HMM, the computational effort remains essentially unchanged. For this modified version we also prove convergence to the homogenized wave equation, but in contrast to the original FE-HMM the long-time dispersive behavior is recovered. The convergence proofs for the FE-HMM follow from new Strang-type results for the wave equation. The results are general enough such that the FE-HMM with and without the long-time correction fits into the setting, even if numerical quadrature is used to evaluate the arising L^2 inner product. In addition to these results we give alternative formulations of the FE-HMM, where the elliptic micro problems are replaced by hyperbolic ones. All the results are supported by numerical tests. The versatility of the method is demonstrated by various numerical examples

Some Applications of the Generalized Multiscale Finite Element Method

Many materials in nature are highly heterogeneous and their properties can vary at different scales. Direct numerical simulations in such multiscale media are prohibitively expensive and some types of model reduction are needed. Typical model reduction techniques include upscaling and multiscale methods. In upscaling methods, one upscales the multiscale media properties so that the problem can be solved on a coarse grid. In multiscale method, one constructs multiscale basis functions that capture media information and solves the problem on the coarse grid. Generalized Multiscale Finite Element Method (GMsFEM) is a recently proposed model reduction technique and has been used for various practical applications. This method has no assumption about the media properties, which can have any type of complicated structure. In GMsFEM, we first create a snapshot space, and then solve a carefully chosen eigenvalue problem to form the offline space. One can also construct online space for the parameter dependent problems. It is shown theoretically and numerically that the GMsFEM is very efficient for the heterogeneous problems involving high-contrast, no-scale separation. In this dissertation, we apply the GMsFEM to perform model reduction for the steady state elasticity equations in highly heterogeneous media though some of our applications are motivated by elastic wave propagation in subsurface. We will consider three kinds of coupling mechanism for different situations. For more practical purposes, we will also study the applications of the GMsFEM for the frequency domain acoustic wave equation and the Reverse Time Migration (RTM) based on the time domain acoustic wave equation