Summary: Variational Theory and Domain Decomposition for Nonlocal Problems
, 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
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.