34 research outputs found
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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
High performance implementation of 3D FEM for nonlocal Poisson problem with different ball approximation strategies
Nonlocality brings many challenges to the implementation of finite element
methods (FEM) for nonlocal problems, such as large number of queries and invoke
operations on the meshes. Besides, the interactions are usually limited to
Euclidean balls, so direct numerical integrals often introduce numerical
errors. The issues of interactions between the ball and finite elements have to
be carefully dealt with, such as using ball approximation strategies. In this
paper, an efficient representation and construction methods for approximate
balls are presented based on combinatorial map, and an efficient parallel
algorithm is also designed for assembly of nonlocal linear systems.
Specifically, a new ball approximation method based on Monte Carlo integrals,
i.e., the fullcaps method, is also proposed to compute numerical integrals over
the intersection region of an element with the ball
Analysis and Simulations of a Nonlocal Gray-Scott Model
The Gray-Scott model is a set of reaction-diffusion equations that describes
chemical systems far from equilibrium. Interest in this model stems from its
ability to generate spatio-temporal structures, including pulses, spots,
stripes, and self-replicating patterns. We consider an extension of this model
in which the spread of the different chemicals is assumed to be nonlocal, and
can thus be represented by an integral operator. In particular, we focus on the
case of strictly positive, symmetric, convolution kernels that have a
finite second moment. Modeling the equations on a finite interval, we prove the
existence of small-time weak solutions in the case of nonlocal Dirichlet and
Neumann boundary constraints. We then use this result to develop a finite
element numerical scheme that helps us explore the effects of nonlocal
diffusion on the formation of pulse solutions.Comment: 28 pages, 2 figure
Schnelle Löser für partielle Differentialgleichungen
[no abstract available
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal