258 research outputs found
A localized orthogonal decomposition method for semi-linear elliptic problems
In this paper we propose and analyze a new Multiscale Method for solving
semi-linear elliptic problems with heterogeneous and highly variable
coefficient functions. For this purpose we construct a generalized finite
element basis that spans a low dimensional multiscale space. The basis is
assembled by performing localized linear fine-scale computations in small
patches that have a diameter of order H |log H| where H is the coarse mesh
size. Without any assumptions on the type of the oscillations in the
coefficients, we give a rigorous proof for a linear convergence of the H1-error
with respect to the coarse mesh size. To solve the arising equations, we
propose an algorithm that is based on a damped Newton scheme in the multiscale
space
Robust error estimates in weak norms for advection dominated transport problems with rough data
We consider mixing problems in the form of transient convection--diffusion
equations with a velocity vector field with multiscale character and rough
data. We assume that the velocity field has two scales, a coarse scale with
slow spatial variation, which is responsible for advective transport and a fine
scale with small amplitude that contributes to the mixing. For this problem we
consider the estimation of filtered error quantities for solutions computed
using a finite element method with symmetric stabilization. A posteriori error
estimates and a priori error estimates are derived using the multiscale
decomposition of the advective velocity to improve stability. All estimates are
independent both of the P\'eclet number and of the regularity of the exact
solution
Reactive Flow and Transport Through Complex Systems
The meeting focused on mathematical aspects of reactive flow, diffusion and transport through complex systems. The research interest of the participants varied from physical modeling using PDEs, mathematical modeling using upscaling and homogenization, numerical analysis of PDEs describing reactive transport, PDEs from fluid mechanics, computational methods for random media and computational multiscale methods
On the periodic homogenization of elliptic equations in non-divergence form with large drifts
We study the quantitative homogenization of linear second order elliptic
equations in non-divergence form with highly oscillating periodic diffusion
coefficients and with large drifts, in the so-called ``centered'' setting where
homogenization occurs and the large drifts contribute to the effective
diffusivity. Using the centering condition and the invariant measures
associated to the underlying diffusion process, we transform the equation into
divergence form with modified diffusion coefficients but without drift. The
latter is in the standard setting for which quantitative homogenization results
have been developed systematically. An application of those results then yields
quantitative estimates, such as the convergence rates and uniform Lipschitz
regularity, for equations in non-divergence form with large drifts.Comment: 16 page
A learning-based multiscale model for reactive flow in porous media
We study solute-laden flow through permeable geological formations with a
focus on advection-dominated transport and volume reactions. As the fluid flows
through the permeable medium, it reacts with the medium, thereby changing the
morphology and properties of the medium; this in turn, affects the flow
conditions and chemistry. These phenomena occur at various lengths and time
scales, and makes the problem extremely complex. Multiscale modeling addresses
this complexity by dividing the problem into those at individual scales, and
systematically passing information from one scale to another. However, accurate
implementation of these multiscale methods are still prohibitively expensive.
We present a methodology to overcome this challenge that is computationally
efficient and quantitatively accurate. We introduce a surrogate for the
solution operator of the lower scale problem in the form of a recurrent neural
operator, train it using one-time off-line data generated by repeated solutions
of the lower scale problem, and then use this surrogate in application-scale
calculations. The result is the accuracy of concurrent multiscale methods, at a
cost comparable to those of classical models. We study various examples, and
show the efficacy of this method in understanding the evolution of the
morphology, properties and flow conditions over time in geological formations
Discontinuous Galerkin finite element heterogeneous multiscale method for advection-diffusion problems with multiple scales
A discontinuous Galerkin finite element heterogeneous multiscale method is proposed for advectiondiffusion problems with highly oscillatory coefficients. The method is based on a coupling of a discontinuous Galerkin discretization for an effective advection-diffusion problem on a macroscopic mesh, whose a priori unknown data are recovered from micro finite element calculations on sampling domains within each macro element. The computational work involved is independent of the high oscillations in the problem at the smallest scale. The stability of our method (depending on both macro and micro mesh sizes) is established for both diffusion dominated and advection dominated regimes without any assumptions about the type of heterogeneities in the data. Fully discrete a priori error bounds are derived for locally periodic data. Numerical experiments confirm the theoretical error estimates
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