Derek Olson

Derek Olson, Xingjie Li, Christoph Ortner et al.

We formulate the blended force-based quasicontinuum (BQCF) method for multilattices and develop rigorous error estimates in terms of the approximation parameters: atomistic region, blending region and continuum finite element mesh. Balancing the approximation parameters yields a convergent atomistic/continuum multiscale method for multilattices with point defects, including a rigorous convergence rate in terms of the computational cost. The analysis is illustrated with numerical results for a Stone--Wales defect in graphene.

Derek Olson, Soumitra Shukla, Gideon Simpson et al.

We examine the Petviashvilli method for solving the equation $ ϕ- Δϕ= |ϕ|^{p-1} ϕ$ on a bounded domain $Ω\subset \mathbb{R}^d$ with Dirichlet boundary conditions. We prove a local convergence result, using spectral analysis, akin to the result for the problem on $\mathbb{R}$ by Pelinovsky & Stepanyants, 2004. We also prove a global convergence result by generating a suite of nonlinear inequalities for the iteration sequence, and we show that the sequence has a natural energy that decreases along the sequence.

Derek Olson, Alexander V. Shapeev, Pavel Bochev et al.

We formulate and analyze an optimization-based Atomistic-to-Continuum (AtC) coupling method for problems with point defects. Near the defect core the method employs a potential-based atomistic model, which enables accurate simulation of the defect. Away from the core, where site energies become nearly independent of the lattice position, the method switches to a more efficient continuum model. The two models are merged by minimizing the mismatch of their states on an overlap region, subject to the atomistic and continuum force balance equations acting independently in their domains. We prove that the optimization problem is well-posed and establish error estimates.