503 research outputs found
Robust numerical upscaling of elliptic multiscale problems at high contrast
We present a new approach to the numerical upscaling for elliptic problems
with rough diffusion coefficient at high contrast. It is based on the
localizable orthogonal decomposition of into the image and the kernel of
some novel stable quasi-interpolation operators with local -approximation
properties, independent of the contrast. We identify a set of sufficient
assumptions on these quasi-interpolation operators that guarantee in principle
optimal convergence without pre-asymptotic effects for high-contrast
coefficients. We then give an example of a suitable operator and establish the
assumptions for a particular class of high-contrast coefficients. So far this
is not possible without any pre-asymptotic effects, but the optimal convergence
is independent of the contrast and the asymptotic range is largely improved
over other discretisation schemes. The new framework is sufficiently flexible
to allow also for other choices of quasi-interpolation operators and the
potential for fully robust numerical upscaling at high contrast
Numerical homogenization of elliptic PDEs with similar coefficients
We consider a sequence of elliptic partial differential equations (PDEs) with
different but similar rapidly varying coefficients. Such sequences appear, for
example, in splitting schemes for time-dependent problems (with one coefficient
per time step) and in sample based stochastic integration of outputs from an
elliptic PDE (with one coefficient per sample member). We propose a
parallelizable algorithm based on Petrov-Galerkin localized orthogonal
decomposition (PG-LOD) that adaptively (using computable and theoretically
derived error indicators) recomputes the local corrector problems only where it
improves accuracy. The method is illustrated in detail by an example of a
time-dependent two-pase Darcy flow problem in three dimensions
Optimal Local Multi-scale Basis Functions for Linear Elliptic Equations with Rough Coefficient
This paper addresses a multi-scale finite element method for second order
linear elliptic equations with arbitrarily rough coefficient. We propose a
local oversampling method to construct basis functions that have optimal local
approximation property. Our methodology is based on the compactness of the
solution operator restricted on local regions of the spatial domain, and does
not depend on any scale-separation or periodicity assumption of the
coefficient. We focus on a special type of basis functions that are harmonic on
each element and have optimal approximation property. We first reduce our
problem to approximating the trace of the solution space on each edge of the
underlying mesh, and then achieve this goal through the singular value
decomposition of an oversampling operator. Rigorous error estimates can be
obtained through thresholding in constructing the basis functions. Numerical
results for several problems with multiple spatial scales and high contrast
inclusions are presented to demonstrate the compactness of the local solution
space and the capacity of our method in identifying and exploiting this compact
structure to achieve computational savings
A Generalized Multiscale Finite Element Method for the Brinkman Equation
In this paper we consider the numerical upscaling of the Brinkman equation in
the presence of high-contrast permeability fields. We develop and analyze a
robust and efficient Generalized Multiscale Finite Element Method (GMsFEM) for
the Brinkman model. In the fine grid, we use mixed finite element method with
the velocity and pressure being continuous piecewise quadratic and piecewise
constant finite element spaces, respectively. Using the GMsFEM framework we
construct suitable coarse-scale spaces for the velocity and pressure that yield
a robust mixed GMsFEM. We develop a novel approach to construct a coarse
approximation for the velocity snapshot space and a robust small offline space
for the velocity space. The stability of the mixed GMsFEM and a priori error
estimates are derived. A variety of two-dimensional numerical examples are
presented to illustrate the effectiveness of the algorithm.Comment: 22 page
Computational Multiscale Methods for Linear Poroelasticity with High Contrast
In this work, we employ the Constraint Energy Minimizing Generalized
Multiscale Finite Element Method (CEM-GMsFEM) to solve the problem of linear
heterogeneous poroelasticity with coefficients of high contrast. The proposed
method makes use of the idea of energy minimization with suitable constraints
in order to generate efficient basis functions for the displacement and the
pressure. These basis functions are constructed by solving a class of local
auxiliary optimization problems based on eigenfunctions containing local
information on the heterogeneity. Techniques of oversampling are adapted to
enhance the computational performance. Convergence of first order is shown and
illustrated by a number of numerical tests.Comment: 14 pages, 9 figure
- …