1,606 research outputs found
Multiscale Partition of Unity
We introduce a new Partition of Unity Method for the numerical homogenization
of elliptic partial differential equations with arbitrarily rough coefficients.
We do not restrict to a particular ansatz space or the existence of a finite
element mesh. The method modifies a given partition of unity such that optimal
convergence is achieved independent of oscillation or discontinuities of the
diffusion coefficient. The modification is based on an orthogonal decomposition
of the solution space while preserving the partition of unity property. This
precomputation involves the solution of independent problems on local
subdomains of selectable size. We deduce quantitative error estimates for the
method that account for the chosen amount of localization. Numerical
experiments illustrate the high approximation properties even for 'cheap'
parameter choices.Comment: Proceedings for Seventh International Workshop on Meshfree Methods
for Partial Differential Equations, 18 pages, 3 figure
Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors
This paper addresses the variational multiscale stabilization of standard
finite element methods for linear partial differential equations that exhibit
multiscale features. The stabilization is of Petrov-Galerkin type with a
standard finite element trial space and a problem-dependent test space based on
pre-computed fine-scale correctors. The exponential decay of these correctors
and their localisation to local cell problems is rigorously justified. The
stabilization eliminates scale-dependent pre-asymptotic effects as they appear
for standard finite element discretizations of highly oscillatory problems,
e.g., the poor approximation in homogenization problems or the pollution
effect in high-frequency acoustic scattering
Numerical Methods for Multilattices
Among the efficient numerical methods based on atomistic models, the
quasicontinuum (QC) method has attracted growing interest in recent years. The
QC method was first developed for crystalline materials with Bravais lattice
and was later extended to multilattices (Tadmor et al, 1999). Another existing
numerical approach to modeling multilattices is homogenization. In the present
paper we review the existing numerical methods for multilattices and propose
another concurrent macro-to-micro method in the numerical homogenization
framework. We give a unified mathematical formulation of the new and the
existing methods and show their equivalence. We then consider extensions of the
proposed method to time-dependent problems and to random materials.Comment: 31 page
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