Analysis of the Residual Type and the Recovery Type a Posteriori Error Estimators for a Consistent Atomistic-to-Continuum Coupling Method in 1D
This work provides rigorous error estimation for atomistic-to-continuum methods, which is important for adaptive simulations in materials science, but the results are limited to 1D and specific interaction models.
The paper derives and proves the efficiency of a residual-type a posteriori error estimator for a 1D atomistic-to-continuum coupling method, shows its equivalence to a gradient recovery-type estimator in the continuum region, and proposes a novel hybrid estimator. Numerical experiments demonstrate optimal convergence rates and efficiency factors.
We consider the a posteriori error estimation for an atomistic-to-continuum cou- pling scheme for a generic one-dimensional many-body next-nearest-neighbour interaction model in 1D. We derive and rigorously prove the efficiency of the residual type estimator. We prove the equivalence between the residual type and the gradient recovery type estima- tor in the continuum region and propose a (novel) hybrid a a posteriori error estimator by combining the two types of estimators. Our numerical experiments illustrate the optimal con- vergence rate of the adaptive algorithms using these estimators whose efficiency factors are also presented.