6,821 research outputs found
Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods
In this paper, we discuss a general multiscale model reduction framework
based on multiscale finite element methods. We give a brief overview of related
multiscale methods. Due to page limitations, the overview focuses on a few
related methods and is not intended to be comprehensive. We present a general
adaptive multiscale model reduction framework, the Generalized Multiscale
Finite Element Method. Besides the method's basic outline, we discuss some
important ingredients needed for the method's success. We also discuss several
applications. The proposed method allows performing local model reduction in
the presence of high contrast and no scale separation
A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure
In this paper we formulate and test numerically a fully-coupled discontinuous
Galerkin (DG) method for incompressible two-phase flow with discontinuous
capillary pressure. The spatial discretization uses the symmetric interior
penalty DG formulation with weighted averages and is based on a wetting-phase
potential / capillary potential formulation of the two-phase flow system. After
discretizing in time with diagonally implicit Runge-Kutta schemes the resulting
systems of nonlinear algebraic equations are solved with Newton's method and
the arising systems of linear equations are solved efficiently and in parallel
with an algebraic multigrid method. The new scheme is investigated for various
test problems from the literature and is also compared to a cell-centered
finite volume scheme in terms of accuracy and time to solution. We find that
the method is accurate, robust and efficient. In particular no post-processing
of the DG velocity field is necessary in contrast to results reported by
several authors for decoupled schemes. Moreover, the solver scales well in
parallel and three-dimensional problems with up to nearly 100 million degrees
of freedom per time step have been computed on 1000 processors
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
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