Title:

Deterministic simulation of multibeaded models of dilute polymer solutions

We study the convergence of a nonlinear approximation method introduced in the engineering literature for the numerical solution of a highdimensional FokkerPlanck equation featuring in NavierStokesFokkerPlanck systems that arise in kinetic models of dilute polymers. To do so, we build on the analysis carried out recently by Le~Bris, Leli\`evre and Maday (Const. Approx. 30: 621651, 2009) in the case of Poisson's equation on a rectangular domain in $\mathbb{R}^2$, subject to a homogeneous Dirichlet boundary condition, where they exploited the connection of the approximation method with the greedy algorithms from nonlinear approximation theory explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173187, 1996). We extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le~Bris, Leli\`evre and Maday to the technically more complicated situation of the elliptic FokkerPlanck equation, where the role of the Laplace operator is played out by a highdimensional OrnsteinUhlenbeck operator with unbounded drift, of the kind that appears in FokkerPlanck equations that arise in beadspring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a highdimensional Cartesian product configuration space $\mathsf{D} = D_1 \times \dotsm \times D_N$ contained in $\mathbb{R}^{N d}$, where each set $D_i$, $i=1, \dotsc, N$, is a bounded open ball in $\mathbb{R}^d$, $d = 2, 3$. We exploit detailed information on the spectral properties and elliptic regularity of the OrnsteinUhlenbeck operator to give conditions on the true solution of the FokkerPlanck equation which guarantee certain rates of convergence of the greedy algorithms. We extend the analysis to discretized versions of the greedy algorithms.
