110 research outputs found
Nonlocal criteria for compactness in the space of vector fields
This work presents a set of sufficient conditions that guarantee a compact
inclusion in the function space of vector fields defined on a domain that
is either a bounded domain in or itself. The
criteria are nonlocal and are given with respect to nonlocal interaction
kernels that may not be necessarily radially symmetric. Moreover, these
criteria for vector fields are also different from those given for scalar
fields in that the conditions are based on nonlocal interactions involving only
parts of the components of the vector fields
Minimal positive stencils in meshfree finite difference methods for linear elliptic equations in non-divergence form
We design a monotone meshfree finite difference method for linear elliptic
equations in the non-divergence form on point clouds via a nonlocal relaxation
method. Nonlocal approximations of linear elliptic equations are first
introduced to which a meshfree finite difference method applies. Minimal
positive stencils are obtained through a local -type optimization
procedure that automatically guarantees the stability and, therefore, the
convergence of the meshfree discretization for linear elliptic equations. The
key to the success of the method relies on the existence of positive stencils
for a given point cloud geometry. We provide sufficient conditions for the
existence of positive stencils by finding neighbors within an ellipse (2d) or
ellipsoid (3d) surrounding each interior point, generalizing the study for
Poisson's equation by Seibold in 2008. It is well-known that wide stencils are
in general needed for constructing consistent and monotone finite difference
schemes for linear elliptic equations. Our study improves the known theoretical
results on the existence of positive stencils for linear elliptic equations
when the ellipticity constant becomes small. Numerical algorithms and practical
guidance are provided with an eye on the case of small ellipticity constant. We
present numerical results in 2d and 3d at the end
Nonlocal half-ball vector operators on bounded domains: Poincar\'e inequality and its applications
This work contributes to nonlocal vector calculus as an indispensable
mathematical tool for the study of nonlocal models that arises in a variety of
applications. We define the nonlocal half-ball gradient, divergence and curl
operators with general kernel functions (integrable or fractional type with
finite or infinite supports) and study the associated nonlocal vector
identities. We study the nonlocal function space on bounded domains associated
with zero Dirichlet boundary conditions and the half-ball gradient operator and
show it is a separable Hilbert space with smooth functions dense in it. A major
result is the nonlocal Poincar\'e inequality, based on which a few applications
are discussed, and these include applications to nonlocal convection-diffusion,
nonlocal correspondence model of linear elasticity, and nonlocal Helmholtz
decomposition on bounded domains
A quasinonlocal coupling method for nonlocal and local diffusion models
In this paper, we extend the idea of "geometric reconstruction" to couple a
nonlocal diffusion model directly with the classical local diffusion in one
dimensional space. This new coupling framework removes interfacial
inconsistency, ensures the flux balance, and satisfies energy conservation as
well as the maximum principle, whereas none of existing coupling methods for
nonlocal-to-local coupling satisfies all of these properties. We establish the
well-posedness and provide the stability analysis of the coupling method. We
investigate the difference to the local limiting problem in terms of the
nonlocal interaction range. Furthermore, we propose a first order finite
difference numerical discretization and perform several numerical tests to
confirm the theoretical findings. In particular, we show that the resulting
numerical result is free of artifacts near the boundary of the domain where a
classical local boundary condition is used, together with a coupled fully
nonlocal model in the interior of the domain
Multiscale Modeling, Homogenization and Nonlocal Effects: Mathematical and Computational Issues
In this work, we review the connection between the subjects of homogenization
and nonlocal modeling and discuss the relevant computational issues. By further
exploring this connection, we hope to promote the cross fertilization of ideas
from the different research fronts. We illustrate how homogenization may help
characterizing the nature and the form of nonlocal interactions hypothesized in
nonlocal models. We also offer some perspective on how studies of nonlocality
may help the development of more effective numerical methods for
homogenization
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Nonlocal models with a finite range of nonlocal interactions
Nonlocal phenomena are ubiquitous in nature. The nonlocal models investigated in this thesis use integration in replace of differentiation and provide alternatives to the classical partial differential equations. The nonlocal interaction kernels in the models are assumed to be as general as possible and usually involve finite range of nonlocal interactions. Such settings on one hand allow us to connect nonlocal models with the existing classical models through various asymptotic limits of the modeling parameter, and on the other hand enjoy practical significance especially for multiscale modeling and simulations.
To make connections with classical models at the discrete level, the central theme of the numerical analysis for nonlocal models in this thesis concerns with numerical schemes that are robust under the changes of modeling parameters, with mathematical analysis provided as theoretical foundations. Together with extensive discussions of linear nonlocal diffusion and nonlocal mechanics models, we also touch upon other topics such as high order nonlocal models, nonlinear nonlocal fracture models and coupling of models characterized by different scales
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