Regularity analyses and approximation of nonlocal variational equality and inequality problems

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:478

Article

Burkovska, O.¹  Gunzburger, M.¹ 

[1] Florida State Univ, Dept Sci Comp, Dirac Sci Lib 400, Tallahassee, FL 32306 USA

关键词: Nonlocal diffusion;    Nonlocal operator;    Fractional Laplacian;    Variational inequalities;    Regularity of the solution;    Finite. elements;   

DOI  :  10.1016/j.jmaa.2019.05.064

来源: Elsevier

【 摘 要 】

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 types 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. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free