Operator-splitting finite element algorithms for computations of high-dimensional parabolic problems

Keywords: Operator-splitting method Finite element method Parabolic equations High-dimensional problems a b s t r a c t An operator-splitting finite element method for solving high-dimensional parabolic equations is presented. The stability and the error estimates are derived for the proposed numerical scheme. Furthermore, two variants of fully-practical operator-splitting finite element algorithms based on the quadrature points and the nodal points, respectively, are presented. Both the quadrature and the nodal point based operator-splitting algorithms are validated using a three-dimensional (3D) test problem. The numerical results obtained with the full 3D computations and the operator-split 2D + 1D computations are found to be in a good agreement with the analytical solution. Further, the optimal order of convergence is obtained in both variants of the operator-splitting algorithms. The numerical solution of partial differential equations in many mathematical models depends not only on time and space but also on some other properties of the considered problem. For instance, the population balance equation (PBE) in a population balance system with one internal coordinate depends on the time, the physical space and a property of the particles, e.g., the size of the particles. Since the PBE contains also derivatives with respect to the properties of the particles , it is posed on a high-dimensional domain in comparison to all other equations in the population balance system [4,12,13,15]. Another more challenging example is the FENE Fokker–Planck equation modeling polymeric fluids [14]. In fact, for a flow domain contained in R d , the polymer configuration domain for a beadspring chain polymer model consisting of M þ 1 beads and M springs, the configuration space domain is contained in R Md and therefore the Fokker–Planck equation is posed on a domain in R dþMd. Even in the simplest case M ¼ 1 the Fokker–Planck equation has to be solved on a domain in R 2d , i.e., four dimensions when d ¼ 2 and six dimensions when d ¼ 3. The numerical solution of partial differential equations on high-dimensional domains are more challenging due to higher storage requirements and computational complexity. To overcome these challenges the sparse grid method [3,9,10] can be used. In the sparse grid method, a high-dimensional equation can be solved by constructing a tensor product sparse grid space using an one-dimensional multilevel basis for each coordinate direction. Similarly, in the space-time sparse grid method , the sparse grid spaces are constructed by a …