Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Variational Theory and Domain Decomposition for Nonlocal Problems   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 Poincar\'{e} 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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The categories of this paper are: plasm-ph
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The categories of this paper are: supr-con
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The categories of this paper are: adap-org
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The categories of this paper are: solv-int
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The categories of this paper are: chao-dyn
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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: quant-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: atom-ph
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The categories of this paper are: nlin
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The categories of this paper are: q-fin
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The categories of this paper are: gr-qc