245 research outputs found
Regularity and approximation analyses of nonlocal variational equality and inequality problems
We consider linear and obstacle problems driven by a nonlocal integral
operator, for which nonlocal interactions are restricted to a ball of finite
radius. These type of operators are used to model anomalous diffusion and, for
a special choice of the integral kernels, reduce to the fractional Laplace
operator on a bounded domain. By means of a nonlocal vector calculus we recast
the problems in a weak form, leading to corresponding nonlocal variational
equality and inequality problems. We prove optimal regularity results for both
problems, including a higher regularity of the solution and the Lagrange
multiplier. Based on the regularity results, we analyze the convergence of
finite element approximations for a linear problem and illustrate the
theoretical findings by numerical results
Goal-oriented A Posteriori Error Estimation for Finite Volume Methods
A general framework for goal-oriented a posteriori error estimation for
finite volume methods is presented. The framework does not rely on recasting
finite volume methods as special cases of finite element methods, but instead
directly determines error estimators from the discretized finite volume
equations. Thus, the framework can be ap- plied to arbitrary finite volume
methods. It also provides the proper functional settings to address
well-posedness issues for the primal and adjoint problems. Numerical results
are presented to illustrate the validity and effectiveness of the a posteriori
error estimates and their applicability to adaptive mesh refinement
Identification of the diffusion parameter in nonlocal steady diffusion problems
The problem of identifying the diffusion parameter appearing in a nonlocal
steady diffusion equation is considered. The identification problem is
formulated as an optimal control problem having a matching functional as the
objective of the control and the parameter function as the control variable.
The analysis makes use of a nonlocal vector calculus that allows one to define
a variational formulation of the nonlocal problem. In a manner analogous to the
local partial differential equations counterpart, we demonstrate, for certain
kernel functions, the existence of at least one optimal solution in the space
of admissible parameters. We introduce a Galerkin finite element discretization
of the optimal control problem and derive a priori error estimates for the
approximate state and control variables. Using one-dimensional numerical
experiments, we illustrate the theoretical results and show that by using
nonlocal models it is possible to estimate non-smooth and discontinuous
diffusion parameters.Comment: 22 pages, 7 figure
Affine approximation of parametrized kernels and model order reduction for nonlocal and fractional Laplace models
We consider parametrized problems driven by spatially nonlocal integral
operators with parameter-dependent kernels. In particular, kernels with varying
nonlocal interaction radius and fractional Laplace kernels,
parametrized by the fractional power , are studied. In order to
provide an efficient and reliable approximation of the solution for different
values of the parameters, we develop the reduced basis method as a parametric
model order reduction approach. Major difficulties arise since the kernels are
not affine in the parameters, singular, and discontinuous. Moreover, the
spatial regularity of the solutions depends on the varying fractional power
. To address this, we derive regularity and differentiability results with
respect to and , which are of independent interest for other
applications such as optimization and parameter identification. We then use
these results to construct affine approximations of the kernels by local
polynomials. Finally, we certify the method by providing reliable a posteriori
error estimators, which account for all approximation errors, and support the
theoretical findings by numerical experiments
Peridynamics and Material Interfaces
The convergence of a peridynamic model for solid mechanics inside
heterogeneous media in the limit of vanishing nonlocality is analyzed. It is
shown that the operator of linear peridynamics for an isotropic heterogeneous
medium converges to the corresponding operator of linear elasticity when the
material properties are sufficiently regular. On the other hand, when the
material properties are discontinuous, i.e., when material interfaces are
present, it is shown that the operator of linear peridynamics diverges, in the
limit of vanishing nonlocality, at material interfaces. Nonlocal interface
conditions, whose local limit implies the classical interface conditions of
elasticity, are then developed and discussed. A peridynamics material interface
model is introduced which generalizes the classical interface model of
elasticity. The model consists of a new peridynamics operator along with
nonlocal interface conditions. The new peridynamics interface model converges
to the classical interface model of linear elasticity
Optimal Point Sets for Total Degree Polynomial Interpolation in Moderate Dimensions
This paper is concerned with Lagrange interpolation by total degree
polynomials in moderate dimensions. In particular, we are interested in
characterising the optimal choice of points for the interpolation problem,
where we define the optimal interpolation points as those which minimise the
Lebesgue constant. We give a novel algorithm for numerically computing the
location of the optimal points, which is independent of the shape of the domain
and does not require computations with Vandermonde matrices. We perform a
numerical study of the growth of the minimal Lebesgue constant with respect to
the degree of the polynomials and the dimension, and report the lowest values
known as yet of the Lebesgue constant in the unit cube and the unit ball in up
to 10 dimensions
An efficient algorithm for simulating ensembles of parameterized flow problems
Many applications of computational fluid dynamics require multiple
simulations of a flow under different input conditions. In this paper, a
numerical algorithm is developed to efficiently determine a set of such
simulations in which the individually independent members of the set are
subject to different viscosity coefficients, initial conditions, and/or body
forces. The proposed scheme applied to the flow ensemble leads to need to solve
a single linear system with multiple right-hand sides, and thus is
computationally more efficient than solving for all the simulations separately.
We show that the scheme is nonlinearly and long-term stable under certain
conditions on the time-step size and a parameter deviation ratio. Rigorous
numerical error estimate shows the scheme is of first-order accuracy in time
and optimally accurate in space. Several numerical experiments are presented to
illustrate the theoretical results.Comment: 20 pages, 3 figure
A second-order time-stepping scheme for simulating ensembles of parameterized flow problems
We consider settings for which one needs to perform multiple flow simulations
based on the Navier-Stokes equations, each having different values for the
physical parameters and/or different initial condition data, boundary
conditions data, and/or forcing functions. For such settings, we propose a
second-order time accurate ensemble-based method that to simulate the whole set
of solutions, requires, at each time step, the solution of only a single linear
system with multiple right-hand-side vectors. Rigorous analyses are given
proving the conditional stability and error estimates for the proposed
algorithm. Numerical experiments are provided that illustrate the analyses.Comment: arXiv admin note: text overlap with arXiv:1705.0935
A Generalized Nonlocal Calculus with Application to the Peridynamics Model for Solid Mechanics
A nonlocal vector calculus was introduced in [2] that has proved useful for
the analysis of the peridynamics model of nonlocal mechanics and nonlocal
diffusion models. A generalization is developed that provides a more general
setting for the nonlocal vector calculus that is independent of particular
nonlocal models. It is shown that general nonlocal calculus operators are
integral operators with specific integral kernels. General nonlocal calculus
properties are developed, including nonlocal integration by parts formula and
Green's identities. The nonlocal vector calculus introduced in [2] is shown to
be recoverable from the general formulation as a special example. This special
nonlocal vector calculus is used to reformulate the peridynamics equation of
motion in terms of the nonlocal gradient operator and its adjoint. A new
example of nonlocal vector calculus operators is introduced, which shows the
potential use of the general formulation for general nonlocal models
Sharp convergence rates of time discretization for stochastic time-fractional PDEs subject to additive space-time white noise
The stochastic time-fractional equation with space-time white noise
is discretized in time by a backward-Euler convolution quadrature for
which the sharp-order error estimate is
established for , where denotes the spatial dimension,
the approximate solution at the time step, and
the expectation operator. In particular, the result indicates
optimal convergence rates of numerical solutions for both stochastic
subdiffusion and diffusion-wave problems in one spatial dimension. Numerical
examples are presented to illustrate the theoretical analysis
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