TY - JOUR
T1 - Mathematical Framework for Lattice Problems
AU - Guo , B.
AU - Babuška , I.
JO - International Journal of Numerical Analysis and Modeling
VL - 3-4
SP - 307
EP - 341
PY - 2007
DA - 2007/04
SN - 4
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/864.html
KW - lattice, cell, multi-scale, periodic structures, grids, absolute term, linear interpolation, Fourier transform.
AB - <p style="text-align: justify;">We propose a mathematical framework to effectively study
lattice materials with periodic and non-periodic structures over entire
spaces in one, two, and three dimensions. The existence and uniqueness of
solutions for periodic lattice problems with absolute terms are proved
in discrete Sobolev spaces. By Fourier transform discrete lattice
problems are converted to semi-discrete problems for which similar
results are established in semi-discrete Sobolev spaces. For lattice
problems without absolute terms, additional conditions are imposed on
data for the existence and uniqueness of solutions in discrete energy
spaces in one, two and three dimensions. Two concrete examples are
analyzed in the proposed mathematical framework. The mathematical
framework, methodology and techniques in this paper can be utilized or
generalized to non-periodic lattices on entire spaces and boundary value
problems on lattices.</p>