Regularity and approximation analyses of nonlocal variational equality and inequality problems
Provides theoretical foundations for numerical methods in nonlocal models used in anomalous diffusion, but is incremental as it extends known techniques to a specific operator class.
The paper proves optimal regularity results for nonlocal variational equality and inequality problems with finite-range interactions, and uses these to analyze convergence of finite element approximations, supported by numerical results.
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.