Variational theory and domain decomposition for nonlocal problems |
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Authors: | Burak Aksoylu Michael L Parks |
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Institution: | a TOBB University of Economics and Technology, Department of Mathematics, Ankara 06560, Turkey b Louisiana State University, Department of Mathematics, Baton Rouge, LA 70803-4918, USA c Sandia National Laboratories, Applied Mathematics and Applications, P.O. Box 5800, MS 1320, Albuquerque, NM 87185-1320, USA |
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Abstract: | In this article we present the first results on domain decomposition methods for nonlocal operators. We present a nonlocal variational formulation for these operators and establish the well-posedness of associated boundary value problems, proving a nonlocal Poincaré inequality. To determine the conditioning of the discretized operator, we prove a spectral equivalence which leads to a mesh size independent upper bound for the condition number of the stiffness matrix. We then introduce a nonlocal two-domain variational formulation utilizing nonlocal transmission conditions, and prove equivalence with the single-domain formulation. A nonlocal Schur complement is introduced. We establish condition number bounds for the nonlocal stiffness and Schur complement matrices. Supporting numerical experiments demonstrating the conditioning of the nonlocal one- and two-domain problems are presented. |
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Keywords: | Domain decomposition Nonlocal substructuring Nonlocal operators Nonlocal Poincaré inequality p-Laplacian Peridynamics Nonlocal Schur complement Condition number |
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