Numerical Analysis of Nonlocal Problems

In this work, several nonlocal problems are studied. Analysis and computation have been done for these problems. Firstly, we consider the time-dependent nonlocal diffusion and wave equations, formulated in the peridynamics setting. Initial and boundary data are given. For nonlocal diffusion equation, the time derivative is approximated using either an explicit Forward Euler, or implicit Backward Euler scheme. For nonlocal wave equation, we get the dispersion relations and use the Newmark method to discretize the equation. We have reformulated the standard time-step stability conditions, in light of the peridynamics formulation. Also we have obtained convergence results. Secondly, we consider the space-time fractional diffusion equation which is used to model anomalous diffusion in physics. Finite difference, finite element and other methods are used to solve it. For finite difference method, the stability of the numerical schemes is well studied. However, for finite element method, we have not found the results for the stability of the θ schemes, especially for the explicit scheme. Here we get the stability and convergence results for all schemes with 0 ≤ θ ≤ 1. Thirdly, an obstacle problem for a nonlocal operator equation 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, 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 uniqueness 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 the nonlocal and local obstacle problems. Then we use sparse grid collocation, reduced basis and simplified reduced basis methods to solve nonlocal diffusion equation with random input data. Regularity of the solution and the convergence results for numerical methods are proved. The efficiency of these methods for solving the problem is investigated. As the radius of the spatial interaction zone changes, the computation cost varies due to the density of the stiffness matrix. This is quite different from local problems. Finally, the 1-d nonlocal diffusion equation is solved by a continuous piecewise-linear collocation method using a uniform mesh. The time derivative is approximated using any of forward Euler, backward Euler, or Crank-Nicolson scheme. By developing a technique to deal with the singular integral, we are able to extend the method so that its validity is extended to include the case 1/2 ≤ s [less than] 1. We also derive stability conditions and convergence rates.