Analysis and approximation of a nonlocal obstacle problem

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• 作者：Qingguang Guan ; ^{qg11b@my.fsu.edu" class="auth_mail" title="E-mail the corresponding author} ; Max Gunzburger ^{mgunzburger@fsu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Obstacle problems ; Nonlocal variational inequality ; Fractional Laplacian ; Discontinuous ; Well posedness ; Finite element methods
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：313
• 期：Complete
• 页码：102-118
• 全文大小：858 K

文摘

An obstacle problem for a nonlocal operator 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 considered here, 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 local, i.e., partial differential equation, and nonlocal obstacle problems.