6 research outputs found
Mixed GMsFEM for the simulation of waves in highly heterogeneous media
Numerical simulations of waves in highly heterogeneous media have important
applications, but direct computations are prohibitively expensive. In this
paper, we develop a new generalized multiscale finite element method with the
aim of simulating waves at a much lower cost. Our method is based on a mixed
Galerkin type method with carefully designed basis functions that can capture
various scales in the solution. The basis functions are constructed based on
some local snapshot spaces and local spectral problems defined on them. The
spectral problems give a natural ordering of the basis functions in the
snapshot space and allow systematically enrichment of basis functions. In
addition, by using a staggered coarse mesh, our method is energy conserving and
has block diagonal mass matrix, which are desirable properties for wave
propagation. We will prove that our method has spectral convergence, and
present numerical results to show the performance of the method
Computational multiscale methods for first-order wave equation using mixed CEM-GMsFEM
In this paper, we consider a pressure-velocity formulation of the
heterogeneous wave equation and employ the constraint energy minimizing
generalized multiscale finite element method (CEM-GMsFEM) to solve this
problem. The proposed method provides a flexible framework to construct crucial
multiscale basis functions for approximating the pressure and velocity. These
basis functions are constructed by solving a class of local auxiliary
optimization problems over the eigenspaces that contain local information on
the heterogeneity. Techniques of oversampling are adapted to enhance the
computational performance. The first-order convergence of the proposed method
is proved and illustrated by several numerical tests.Comment: 18 pages, 5 figures, proof of Theorem 4.2 modifie
A ray-based IPDG method for high-frequency time-domain acoustic wave propagation in inhomogeneous media
The numerical approximation of high-frequency wave propagation in
inhomogeneous media is a challenging problem. In particular, computing
high-frequency solutions by direct simulations requires several points per
wavelength for stability and usually requires many points per wavelength for a
satisfactory accuracy. In this paper, we propose a new method for the acoustic
wave equation in inhomogeneous media in the time domain to achieve superior
accuracy and stability without using a large number of unknowns. The method is
based on a discontinuous Galerkin discretization together with carefully chosen
basis functions. To obtain the basis functions, we use the idea from
geometrical optics and construct the basis functions by using the leading order
term in the asymptotic expansion. Also, we use a wavefront tracking method and
a dimension reduction procedure to obtain dominant rays in each cell. We show
numerically that the accuracy of the numerical solutions computed by our method
is significantly higher than that computed by the IPDG method using
polynomials.Moreover, the relative errors of our method grow only moderately as
the frequency increases
Constraint energy minimization generalized multiscale finite element method in mixed formulation for parabolic equations
In this paper, we develop the constraint energy minimization generalized
multiscale finite element method (CEM-GMsFEM) in mixed formulation applied to
parabolic equations with heterogeneous diffusion coefficients. The construction
of the method is based on two multiscale spaces: pressure multiscale space and
velocity multiscale space. The pressure space is constructed via a set of
well-designed local spectral problems, which can be solved independently. Based
on the computed pressure multiscale space, we will construct the velocity
multiscale space by applying constrained energy minimization. The convergence
of the proposed method is proved.In particular, we prove that the convergence
of the method depends only on the coarse grid size, and is independent of the
heterogeneities and contrast of thediffusion coefficient. Four typical types of
permeability fields are exploited in the numerical simulations, and the results
indicate that our proposed method works well and gives efficient and accurate
numerical solutions.Comment: 25 page
Explicit and Energy-Conserving Constraint Energy Minimizing Generalized Multiscale Discontinuous Galerkin Method for Wave Propagation in Heterogeneous Media
In this work, we propose a local multiscale model reduction approach for the
time-domain scalar wave equation in a heterogenous media. A fine mesh is used
to capture the heterogeneities of the coefficient field, and the equation is
solved globally on a coarse mesh in the discontinuous Galerkin discretization
setting. The main idea of the model reduction approach is to extract dominant
modes in local spectral problems for representation of important features,
construct multiscale basis functions in coarse oversampled regions by
constraint energy minimization problems, and perform a Petrov-Galerkin
projection and a symmetrization onto the coarse grid. The method is expicit and
energy conserving, and exhibits both coarse-mesh and spectral convergence,
provided that the oversampling size is appropriately chosen. We study the
stability and convergence of our method. We also present numerical results on
the Marmousi model in order to test the performance of the method and verify
the theoretical results.Comment: arXiv admin note: substantial text overlap with arXiv:1909.1246
Iterative Oversampling Technique for Constraint Energy Minimizing Generalized Multiscale Finite Element Method in the Mixed Formulation
In this paper, we develop an iterative scheme to construct multiscale basis
functions within the framework of the Constraint Energy Minimizing Generalized
Multiscale Finite Element Method (CEM-GMsFEM) for the mixed formulation. The
iterative procedure starts with the construction of an energy minimizing
snapshot space that can be used for approximating the solution of the model
problem. A spectral decomposition is then performed on the snapshot space to
form global multiscale space. Under this setting, each global multiscale basis
function can be split into a non-decaying and a decaying parts. The
non-decaying part of a global basis is localized and it is fixed during the
iteration. Then, one can approximate the decaying part via a modified
Richardson scheme with an appropriately defined preconditioner. Using this set
of iterative-based multiscale basis functions, first-order convergence with
respect to the coarse mesh size can be shown if sufficiently many times of
iterations with regularization parameter being in an appropriate range are
performed. Numerical results are presented to illustrate the effectiveness and
efficiency of the proposed computational multiscale method