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arXiv · 1804.10282

Regularity and approximation analyses of nonlocal variational equality and inequality problems

Olena Burkovska·Max Gunzburger
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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.

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Olena Burkovska, Max Gunzburger. 2018-04-26. Regularity and approximation analyses of nonlocal variational equality and inequality problems. https://arxiv.org/abs/1804.10282

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