28 research outputs found
Generalized Multiscale Finite Element Methods for problems in perforated heterogeneous domains
Complex processes in perforated domains occur in many real-world
applications. These problems are typically characterized by physical processes
in domains with multiple scales (see Figure 1 for the illustration of a
perforated domain). Moreover, these problems are intrinsically multiscale and
their discretizations can yield very large linear or nonlinear systems. In this
paper, we investigate multiscale approaches that attempt to solve such problems
on a coarse grid by constructing multiscale basis functions in each coarse
grid, where the coarse grid can contain many perforations. In particular, we
are interested in cases when there is no scale separation and the perforations
can have different sizes. In this regard, we mention some earlier pioneering
works [14, 18, 17], where the authors develop multiscale finite element
methods. In our paper, we follow Generalized Multiscale Finite Element Method
(GMsFEM) and develop a multiscale procedure where we identify multiscale basis
functions in each coarse block using snapshot space and local spectral
problems. We show that with a few basis functions in each coarse block, one can
accurately approximate the solution, where each coarse block can contain many
small inclusions. We apply our general concept to (1) Laplace equation in
perforated domain; (2) elasticity equation in perforated domain; and (3) Stokes
equations in perforated domain. Numerical results are presented for these
problems using two types of heterogeneous perforated domains. The analysis of
the proposed methods will be presented elsewhere
Asymptotic analysis of a semi-linear elliptic system in perforated domains: well-posedness and correctors for the homogenization limit
In this study, we prove results on the weak solvability and homogenization of
a microscopic semi-linear elliptic system posed in perforated media. The model
presented here explores the interplay between stationary diffusion and both
surface and volume chemical reactions in porous media. Our interest lies in
deriving homogenization limits (upscaling) for alike systems and particularly
in justifying rigorously the obtained averaged descriptions. Essentially, we
prove the well-posedness of the microscopic problem ensuring also the
positivity and boundedness of the involved concentrations and then use the
structure of the two scale expansions to derive corrector estimates
delimitating this way the convergence rate of the asymptotic approximates to
the macroscopic limit concentrations. Our techniques include Moser-like
iteration techniques, a variational formulation, two-scale asymptotic
expansions as well as energy-like estimates.Comment: 22 pages, 1 figur
A high-order corrector estimate for a semi-linear elliptic system in perforated domains
We derive in this note a high-order corrector estimate for the homogenization
of a microscopic semi-linear elliptic system posed in perforated domains. The
major challenges are the presence of nonlinear volume and surface reaction
rates. This type of correctors justifies mathematically the convergence rate of
formal asymptotic expansions for the two-scale homogenization settings. As main
tool, we follow the standard approach by the energy-like method to investigate
the error estimate between the micro and macro concentrations and micro and
macro concentration gradients. This work aims at generalizing the results
reported in [2, 7].Comment: 6 pages, 1 figur
Partially Explicit Time Discretization for Nonlinear Time Fractional Diffusion Equations
Nonlinear time fractional partial differential equations are widely used in
modeling and simulations. In many applications, there are high contrast changes
in media properties. For solving these problems, one often uses coarse spatial
grid for spatial resolution. For temporal discretization, implicit methods are
often used. For implicit methods, though the time step can be relatively large,
the equations are difficult to compute due to the nonlinearity and the fact
that one deals with large-scale systems. On the other hand, the discrete system
in explicit methods are easier to compute but it requires small time steps. In
this work, we propose the partially explicit scheme following earlier works on
developing partially explicit methods for nonlinear diffusion equations. In
this scheme, the diffusion term is treated partially explicitly and the
reaction term is treated fully explicitly. With the appropriate construction of
spaces and stability analysis, we find that the required time step in our
proposed scheme scales as the coarse mesh size, which creates a great saving in
computing. The main novelty of this work is the extension of our earlier works
for diffusion equations to time fractional diffusion equations. For the case of
fractional diffusion equations, the constraints on time steps are more severe
and the proposed methods alleviate this since the time step in partially
explicit method scales as the coarse mesh size. We present stability results.
Numerical results are presented where we compare our proposed partially
explicit methods with a fully implicit approach. We show that our proposed
approach provides similar results, while treating many degrees of freedom in
nonlinear terms explicitly.Comment: 20 pages, 15 figure