Localized Orthogonal Decomposition for two-scale Helmholtz-type problems

In this paper, we present a Localized Orthogonal Decomposition (LOD) in Petrov-Galerkin formulation for a two-scale Helmholtz-type problem. The two-scale problem is, for instance, motivated from the homogenization of the Helmholtz equation with high contrast, studied together with a corresponding multiscale method in (Ohlberger, Verf\"urth. A new Heterogeneous Multiscale Method for the Helmholtz equation with high contrast, arXiv:1605.03400, 2016). There, an unavoidable resolution condition on the mesh sizes in terms of the wave number has been observed, which is known as "pollution effect" in the finite element literature. Following ideas of (Gallistl, Peterseim. Comput. Methods Appl. Mech. Engrg. 295:1-17, 2015), we use standard finite element functions for the trial space, whereas the test functions are enriched by solutions of subscale problems (solved on a finer grid) on local patches. Provided that the oversampling parameter $m$, which indicates the size of the patches, is coupled logarithmically to the wave number, we obtain a quasi-optimal method under a reasonable resolution of a few degrees of freedom per wave length, thus overcoming the pollution effect. In the two-scale setting, the main challenges for the LOD lie in the coupling of the function spaces and in the periodic boundary conditions.Comment: 20 page

Multiscale Method for Elastic Wave Propagation in the Heterogeneous, Anisotropic Media

Seismic wave simulation in realistic Earth media with full wavefield methods is a fundamental task in geophysical studies. Conventional approaches such as the finite-difference method and the finite-element method solve the wave equation in geological models represented with discrete grids and elements. When the Earth model includes complex heterogeneities at multiple spatial scales, the simulation requires fine discretization and therefore a system with many degrees of freedom, which often exceeds current computational abilities. In this dissertation, I address this problem by proposing new multiscale methods for simulating elastic wave propagation based on previously developed algorithms for solving the elliptic partial differential equations and the acoustic wave equation. The fundamental motivation for developing the multiscale method is that it can solve the wave equation on a coarsely discretized mesh by incorporating the effects of fine-scale medium properties using so-called multiscale basis functions. This can greatly reduce computation time and degrees of freedom compared with conventional methods. I first derive a numerical homogenization method for arbitrarily heterogeneous, anisotropic media that utilizes the multiscale basis functions determined from a local linear elasticity equation to compute effective, anisotropic properties, and these equivalent elastic medium parameters can be used directly in existing elastic modeling algorithms. Then I extend the approach by constructing multiple basis functions using two types of appropriately defined local spectral linear elasticity problems. Given the eigenfunctions determined from local spectral problems, I develop a generalized multiscale finite-element method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media in both continuous Galerkin (CG) and discontinuous Galerkin (DG) formulations. The advantage of the multiscale basis functions is they are model-dependent, unlike the predefined polynomial basis functions applied in conventional finite-element methods. For this reason, the GMsFEM can effectively capture the influence of fine-scale variation of the media. I present results for several numerical experiments to verify the effectiveness of both the numerical homogenization method and GMsFEM. These tests show that the effectiveness of the multiscale method relies on the appropriate choice of boundary conditions that are applied for the local problem in numerical homogenization method and on the selection of basis functions from a large set of eigenfunctions contained in local spectral problems in GMsFEM. I develop methods for solving both these problems, and the results confirm that the multiscale method can be powerful tool for providing accurate full wavefield solutions in heterogeneous, anisotropic media, yet with reduced computation time and degrees of freedom compared with conventional full wavefield modeling methods. Specially, I applied the DG-GMsFEM to the Marmousi-2 elastic model, and find that DG-GMsFEM can greatly reduce the computation time compared with continuous Galerkin (CG) FEM

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

A priori error analysis of the finite element heterogeneous multiscale method for the wave equation in heterogeneous media over long time

A fully discrete a priori analysis of the finite element heterogeneous multiscale method introduced in [A. Abdulle, M. Grote, and C. Stohrer, Multiscale Model. Simul., 12, 2014, pp. 1135-1166] for the wave equation with highly oscillatory coefficients over long time is presented. A sharp a priori convergence rate for the numerical method is derived for long time intervals. The effective model over long time is a Boussinesq-type equation that has been shown to approximate the one-dimensional multiscale wave equation with epsilon-periodic coefficients up to time O(epsilon(-2)) in [A. Lamacz, Math. Models Methods Appl. Sci., 21, 2011, pp. 1871-1899]. In this paper we also revisit this result by deriving and analyzing a family of effective Boussinesq-type equations for the approximation of the multiscale wave equation that depends on the normalization chosen for certain micro functions used to define the macroscopic models

Finite element heterogeneous multiscale method for Elastic Waves in Heterogeneous Media

A finite element heterogeneous multiscale method (FE-HMM) is proposed for the simulation of time-dependent elastic waves in a rapidly varying heterogeneous elastic medium. It is based on a standard finite element discretization of an effective wave equation at the macro scale, whose a priori unknown effective material coefficients are computed on sampling domains at the micro scale within each macro finite element. Hence the computational effort becomes independent of the highly heterogeneous elastic medium at the smallest scale. Optimal error estimates and convergence rates in the energy and the L2 norm are derived, which are explicit in the macro and micro discretization errors. Numerical experiments verify the sharpness of the error bounds and illustrate the versatility of the method for non-periodic, layered or stochastic media