Generalized Multiscale Finite Element Method for the Steady State Linear Boltzmann Equation

Abstract

The Boltzmann equation, as a model equation in statistical mechanics, is used to describe the statistical behavior of a large number of particles driven by the same physics laws. Depending on the media and the particles to be modeled, the equation has slightly different forms. In this article, we investigate a model Boltzmann equation with highly oscillatory media in the small Knudsen number regime, and study the numerical behavior of the Generalized Multi-scale Finite Element Method (GMsFEM) in the fluid regime when high oscillation in the media presents. The Generalized Multi-scale Finite Element Method (GMsFEM) is a general approach [7] to numerically treat equations with multi-scale structures. The method is divided into the offline and online steps. In the offline step, basis functions are prepared from a snapshot space via a well-designed generalized eigenvalue problem (GEP), and these basis functions are then utilized to patch up for a solution through DG formulation in the online step to incorporate specific boundary and source information. We prove the wellposedness of the method on the Boltzmann equation, and show that the GEP formulation provides a set of optimal basis functions that achieve spectral convergence. Such convergence is independent of the oscillation in the media, or the smallness of the Knudsen number, making it one of the few methods that simultaneously achieve numerical homogenization and asymptotic preserving properties across all scales of oscillations and the Knudsen number.

A conservative multiscale method for stochastic highly heterogeneous flow

“…V ms,k : a velocity multiscale space at enrichment iteration k. V ms := V ms,m : a final velocity multiscale space is generated after m enrichment iterations. In particular, u f (x; ω, f ) and p f (x; ω, f ) are velocity and pressure solutions to (19) and (20); u ms (x; ω, f ) and p ms (x; ω, f ) are velocity and pressure solutions to (21) and (22) as follows.…”

“…Recall that κ 1 := κ(x; ω 1 ). Similarly as the proof in Theorem 2, we subtract (19) and (20) corresponding with f 1 and f 2 to get…”

“…However, some important small scale information is hard to be incorporated in the reduced models, which may hinder producing an accurate approximation. Multiscale methods [25,18,23,30,10,38,19], on the other hand, are motivated to capture small-scale information of the underlying heterogeneous media in multiscale bases. Even using only a limited number of multiscale bases, the reduced-order solution can still have relatively good accuracy.…”

Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model

“…Multiscale methods are widely used to solve various problems in domains with complex heterogeneities. Some of the popular methods are the multiscale finite element method(MsFEM) [27,24,28], mixed multiscale finite element method(Mixed MsFEM) [29,30], generalize multiscale finite element method(GMsFEM) [25,31,32,33,34], heterogeneous multiscale methods [35,36,37], multiscale finite volume method (MsFVM) [38,39,40], constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) [41,41] and etc. Recently, in [42,43,44], the authors present a special design of the multiscale basis functions to solve problems in fractured porous media, which obtains the basis functions based on the constrained energy minimization problems and nonlocal multicontinuum (NLMC) method.…”

Constraint Energy Minimizing Generalized Multiscale Finite Element Method for multi-continuum Richards equations

“…A variety of domain decomposition methods [43,56,55] have leveraged the idea of building local basis functions utilizing spectral problem. Recently, the GMsFEM has been successfully applied to a variety of problems [2,86,23,79,81,1,3,5,18,27,26,59].…”