Deterministic Simulation of Multi-Beaded 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 high-dimensional Fokker– Planck equation featuring in Navier–Stokes–Fokker–Planck systems that arise in kinetic models of dilute polymers. To do so, we build on the analysis carried out recently by Le Bris, Lelièvre and Maday (Const. Approx. 30: 621–651, 2009) in the case of Poisson’s equation on a rectangular domain in R2, 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. 5:173–187, 1996). We extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le Bris, Lelièvre and Maday to the technically more complicated situation of the elliptic Fokker–Planck equation, where the role of the Laplace operator is played out by a high-dimensional Ornstein–Uhlenbeck operator with unbounded drift, of the kind that appears in Fokker–Planck equations that arise in bead-spring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a highdimensional Cartesian product configuration space D = D1 × · · · ×DN contained in RNd, where each set Di, i = 1, . . . , N , is a bounded open ball in Rd, d = 2, 3. We exploit detailed information on the spectral properties and elliptic regularity of the Ornstein–Uhlenbeck operator to give conditions on the true solution of the Fokker–Planck equation which guarantee certain rates of convergence of the greedy algorithms. We extend the analysis to discretized versions of the greedy algorithms.