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Variational Theory and Domain Decomposition for Nonlocal Problems Burak Aksoylua,b,1
 

Summary: Variational Theory and Domain Decomposition for Nonlocal Problems
Burak Aksoylua,b,1
, Michael L. Parksc,2,
aTOBB University of Economics and Technology, Department of Mathematics, Ankara, 06560, Turkey
bLouisiana State University, Department of Mathematics, Baton Rouge, LA 70803-4918 USA
cSandia National Laboratories, Applied Mathematics and Applications, P.O. Box 5800, MS 1320, Albuquerque, NM
87185-1320 USA
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 PoincarŽe inequality. To determine the conditioning of the dis-
cretized 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 formula-
tion 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.
Keywords: Domain decomposition, nonlocal substructuring, nonlocal operators, nonlocal PoincarŽe
inequality, p-Laplacian, peridynamics, nonlocal Schur complement, condition number.
1. Introduction

  

Source: Aksoylu, Burak - Center for Computation and Technology & Department of Mathematics, Louisiana State University

 

Collections: Computer Technologies and Information Sciences