Regularity and approximation analyses of nonlocal variational equality and inequality problems
Abstract
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.
Cite
@article{arxiv.1804.10282,
title = {Regularity and approximation analyses of nonlocal variational equality and inequality problems},
author = {Olena Burkovska and Max Gunzburger},
journal= {arXiv preprint arXiv:1804.10282},
year = {2018}
}