ESAIM: Mathematical Modelling and Numerical Analysis

Research Article

The role of the patch test in 2D atomistic-to-continuum coupling methods 

Christoph Ortner

Mathematics Institute, Zeeman Building, University of Warwick, CV4 7AL Coventry, UK.


For a general class of atomistic-to-continuum coupling methods, coupling multi-body interatomic potentials with a P1-finite element discretisation of Cauchy–Born nonlinear elasticity, this paper adresses the question whether patch test consistency (or, absence of ghost forces) implies a first-order error estimate. In two dimensions it is shown that this is indeed true under the following additional technical assumptions: (i) an energy consistency condition, (ii) locality of the interface correction, (iii) volumetric scaling of the interface correction, and (iv) connectedness of the atomistic region. The extent to which these assumptions are necessary is discussed in detail.

(Received March 14 2011)

(Revised October 21 2011)

(Online publication March 27 2012)

Key Words:

  • Atomistic models;
  • atomistic-to-continuum coupling;
  • quasicontinuum method;
  • coarse graining;
  • ghost forces;
  • patch test;
  • consistency

Mathematics Subject Classification:

  • 65N12;
  • 65N15;
  • 70C20


  This work was supported by the EPSRC Critical Mass Programme “New Frontiers in the Mathematics of Solids” (OxMoS) and by the EPSRC grant EP/H003096 “Analysis of atomistic-to-continuum coupling methods”.