Connecting atomistic-to-continuum coupling and domain decomposition. (English) Zbl 1160.65343

Parks, Michael L.; Bochev, Pavel B.; Lehoucq, Richard B.

Multiscale Model. Simul. 7, No. 1, 362-380 (2008).

Summary: Many atomistic/continuum coupling algorithms utilize an overlapping subdomain method, where boundary data for local solves in atomistic and discretized continuum subdomains is provided from local solves in neighboring subdomains. Such coupling algorithms are closely related to the classical alternating Schwarz domain decomposition method, although little to no convergence or error analysis exists for such methods in an atomistic/continuum framework. We consider a specific alternating Schwarz algorithm for coupling a nonlocal atomistic model with a local finite element model and carry out a convergence and error analysis along with supporting numerical experiments.

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55	Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F10	Iterative numerical methods for linear systems
35J25	Boundary value problems for second-order elliptic equations
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs

Keywords:

atomistic-to-continuum; lattice statics; alternating Schwarz domain decomposition method; algorithm; finite element; convergence; error analysis; numerical experiments