Variational theory and domain decomposition for nonlocal problems

Authors

Burak Aksoylu, TOBB University of Economics and Technology, Ankara, Turkey; Louisiana State University
Michael L. Parks, Sandia National Laboratories, Applied Mathematics and Applications, Albuquerque, NM

Date of this Version

2011

Citation

Journal of Applied Mathematics and Computation 217 (2011) 6498–6515; doi:10.1016/j.amc.2011.01.027

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 oneand two-domain problems are presented.