Results on Nonlocal Boundary Value Problems |
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Authors: | Burak Aksoylu Tadele Mengesha |
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Institution: | 1. Department of Mathematics , TOBB University of Economics and Technology , Ankara, Turkey;2. Department of Mathematics , Louisiana State University , Baton Rouge, Louisiana, USA baksoylu@etu.edu.tr;4. Department of Mathematics , Louisiana State University , Baton Rouge, Louisiana, USA;5. Department of Mathematics and Statistics , Coastal Carolina University , Conway, South Carolina, USA |
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Abstract: | In this article, we provide a variational theory for nonlocal problems where nonlocality arises due to the interaction in a given horizon. With this theory, we prove well-posedness results for the weak formulation of nonlocal boundary value problems with Dirichlet, Neumann, and mixed boundary conditions for a class of kernel functions. The motivating application for nonlocal boundary value problems is the scalar stationary peridynamics equation of motion. The well-posedness results support practical kernel functions used in the peridynamics setting. We also prove a spectral equivalence estimate which leads to a mesh size independent upper bound for the condition number of an underlying discretized operator. This is a fundamental conditioning result that would guide preconditioner construction for nonlocal problems. The estimate is a consequence of a nonlocal Poincaré-type inequality that reveals a horizon size quantification. We provide an example that establishes the sharpness of the upper bound in the spectral equivalence. |
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Keywords: | Condition number Nonlocal boundary value problems Nonlocal operators Nonlocal Poincaré inequality Peridynamics Preconditioning Well-posedness |
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