903 research outputs found
Multiscale Finite Element Methods for Nonlinear Problems and their Applications
In this paper we propose a generalization of multiscale finite element methods (Ms-FEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the over-sampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities
Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast
We construct finite-dimensional approximations of solution spaces of
divergence form operators with -coefficients. Our method does not
rely on concepts of ergodicity or scale-separation, but on the property that
the solution space of these operators is compactly embedded in if source
terms are in the unit ball of instead of the unit ball of .
Approximation spaces are generated by solving elliptic PDEs on localized
sub-domains with source terms corresponding to approximation bases for .
The -error estimates show that -dimensional spaces
with basis elements localized to sub-domains of diameter (with ) result in an
accuracy for elliptic, parabolic and hyperbolic
problems. For high-contrast media, the accuracy of the method is preserved
provided that localized sub-domains contain buffer zones of width
where the contrast of the medium
remains bounded. The proposed method can naturally be generalized to vectorial
equations (such as elasto-dynamics).Comment: Accepted for publication in SIAM MM
Discrete Geometric Structures in Homogenization and Inverse Homogenization with application to EIT
We introduce a new geometric approach for the homogenization and inverse
homogenization of the divergence form elliptic operator with rough conductivity
coefficients in dimension two. We show that conductivity
coefficients are in one-to-one correspondence with divergence-free matrices and
convex functions over the domain . Although homogenization is a
non-linear and non-injective operator when applied directly to conductivity
coefficients, homogenization becomes a linear interpolation operator over
triangulations of when re-expressed using convex functions, and is a
volume averaging operator when re-expressed with divergence-free matrices.
Using optimal weighted Delaunay triangulations for linearly interpolating
convex functions, we obtain an optimally robust homogenization algorithm for
arbitrary rough coefficients. Next, we consider inverse homogenization and show
how to decompose it into a linear ill-posed problem and a well-posed non-linear
problem. We apply this new geometric approach to Electrical Impedance
Tomography (EIT). It is known that the EIT problem admits at most one isotropic
solution. If an isotropic solution exists, we show how to compute it from any
conductivity having the same boundary Dirichlet-to-Neumann map. It is known
that the EIT problem admits a unique (stable with respect to -convergence)
solution in the space of divergence-free matrices. As such we suggest that the
space of convex functions is the natural space in which to parameterize
solutions of the EIT problem
An Equation-Free Approach for Second Order Multiscale Hyperbolic Problems in Non-Divergence Form
The present study concerns the numerical homogenization of second order
hyperbolic equations in non-divergence form, where the model problem includes a
rapidly oscillating coefficient function. These small scales influence the
large scale behavior, hence their effects should be accurately modelled in a
numerical simulation. A direct numerical simulation is prohibitively expensive
since a minimum of two points per wavelength are needed to resolve the small
scales. A multiscale method, under the equation free methodology, is proposed
to approximate the coarse scale behaviour of the exact solution at a cost
independent of the small scales in the problem. We prove convergence rates for
the upscaled quantities in one as well as in multi-dimensional periodic
settings. Moreover, numerical results in one and two dimensions are provided to
support the theory
Algorithmic Monotone Multiscale Finite Volume Methods for Porous Media Flow
Multiscale finite volume methods are known to produce reduced systems with
multipoint stencils which, in turn, could give non-monotone and out-of-bound
solutions. We propose a novel solution to the monotonicity issue of multiscale
methods. The proposed algorithmic monotone (AM- MsFV/MsRSB) framework is based
on an algebraic modification to the original MsFV/MsRSB coarse-scale stencil.
The AM-MsFV/MsRSB guarantees monotonic and within bound solutions without
compromising accuracy for various coarsening ratios; hence, it effectively
addresses the challenge of multiscale methods' sensitivity to coarse grid
partitioning choices. Moreover, by preserving the near null space of the
original operator, the AM-MsRSB showed promising performance when integrated in
iterative formulations using both the control volume and the Galerkin-type
restriction operators. We also propose a new approach to enhance the
performance of MsRSB for MPFA discretized systems, particularly targeting the
construction of the prolongation operator. Results show the potential of our
approach in terms of accuracy of the computed basis functions and the overall
convergence behavior of the multiscale solver while ensuring a monotone
solution at all times.Comment: 29 pages, 20 figure
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