We present novel techniques for obtaining the basic estimates of virtual element methods in terms of the shape regularity of polygonal/polyhedral meshes. We also derive new error estimates for the Poisson problem in two and three dimensions.

A time‐dependent nonlocal wave equation is considered. A feature of the model is that instead of boundary conditions, constraints over regions having finite measures are imposed. The Newmark scheme is considered for discretizing the time derivative and piecewise‐linear finite element methods are used for spatial discretization. For certain ranges of a parameter appearing in the Newmark scheme, unconditional stability is proved; in particular, this result applies to the backward‐Euler‐like and Crank‐Nicolson‐like schemes. For other values of the parameter which includes the forward‐Euler‐like scheme, conditional stability is proved. Dispersion relations for the nonlocal wave equation in one and two dimensions are derived. Comparisons with the analogous results for the classical wave equation are provided as the results of numerical experiments that illustrate the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 500–516, 2015

No abstract

An obstacle problem for a nonlocal operator is considered; the operator is a nonlocal integral analogue of the Laplacian operator and, as a special case, reduces to the fractional Laplacian. In the analysis of classical obstacle problems for the Laplacian, the obstacle is taken to be a smooth function. For the nonlocal obstacle problem considered here, obstacles are allowed to have jump discontinuities. We cast the nonlocal obstacle problem as a minimization problem wherein the solution is constrained to lie above the obstacle. We prove the existence and unique of a solution in an appropriate function space. Then, the well posedness and convergence of finite element approximations are demonstrated. The results of numerical experiments are provided that illustrate the theoretical results and the differences between solutions of local, i.e., partial differential equation, and nonlocal obstacle problems.

The construction, analysis, and application of reduced-basis methods for uncertainty quantification problems involving nonlocal diffusion problems with random input data is the subject of this work. Because of the lack of sparsity of discretized nonlocal models relative to analogous local partial differential equation models, the need for reducedorder modeling is much more acute in the nonlocal setting. In this effort, we develop reduced-basis approximations for nonlocal diffusion equations with affine random coefficients. Efficiency estimates of the proposed greedy reduced-basis methods is provided. Numerical examples are used to illustrate the effect varying various model parameters have on the efficiency and accuracy of the reduced-basis method relative to the sparse grid interpolation using the full finite element method. It is shown that the proposed reduced-basis approach can indeed provide substantial savings over the standard sparse grid method when combined with full finite element solvers. is used to describe long-range spatial interactions. For example, in solid mechanics, the strain energy density at a point x can depend on points in a neighborhood of x having non-zero volume. This is in contrast to local PDE models where interactions occur only in an infinitesimal neighborhood of the point x. Further, without some interventions, PDEs cannot model discontinuous behavior such as crack nucleation and propagation. Nonlocal models that are free of spatial derivatives can gracefully handle continuous and discontinuous behaviors such as fracture [1,2]. In image denosing, it is often reasonable to assume that the noise at any point is related directly to that any other point in the image, or at least to points in relevant regions, rather than just its immediate neighbors. This leads to the study of nonlocal operators in image analyses [3][4][5][6] that, similar to the nonlocal model in continuum mechanics, can process both structures (continuous) and textures (discontinuous) within the same framework. In the case considered here, i.e., diffusion, nonlocal interaction give rise to anomalous diffusion, i.e., to diffusive behavior that does not follow Fickian laws or that can be modeled by Brownian motions. In [10], a finite element framework and a thorough variational analysis is provided for nonlocal problems based on the nonlocal vector calculus recently developed in [11].On the other hand, the inputs to a model, be they local or nonlocal, are often subject to uncertainties. For example, in the PDE setting, coefficients, forcing terms, and data in initial and boundary conditions may be uncertain. Thus, uncertainty quantification, i.e., the analysis and quantification of the uncertainties in the outputs of a model due to uncertainties in its inputs, has been of growing interest in recent years. For local problems modeled by differential equations with random input data, the classical Monte Carlo method is widely used. The appealing properties of this method are its ease of implementation based on existing det...