Stability, Instability, and Error of the Force-based Quasicontinuum Approximation
M. Dobson, M. Luskin and C. Ortner
Archive for Rational Mechanics and Analysis, 197, 179-202 (2010).
ABSTRACT
Due to their algorithmic simplicity and high accuracy, force-based model
coupling techniques are popular tools in computational physics. For example, the
force-based quasicontinuum (QCF) approximation is the only known pointwise
consistent quasicontinuum approximation for coupling a general atomistic model
with a finite element continuum model. In this paper, we present a detailed
stability and error analysis of this method. Our optimal order error estimates
provide a theoretical justification for the high accuracy of the QCF
approximation: they clearly demonstrate that the computational efficiency of
continuum modeling can be utilized without a significant loss of accuracy if
defects are captured in the atomistic region. The main challenge we need to
overcome is the fact that the linearized QCF operator is typically not positive
definite. Moreover, we prove that no uniform inf-sup stability condition holds
for discrete versions of the W-1,W-p-W-1,W-q "duality pairing" with 1/p + 1/q =
1, if 1 <= p < infinity. However, we were able to establish an inf-sup stability
condition for a discrete version of the W-1,W-infinity-W-1,W-1 "duality pairing"
which leads to optimal order error estimates in a discrete W-1,W-infinity-norm.