An operator-splitting heterogeneous finite element method for population balance equations: stability and convergence

Abstract

We present a heterogeneous finite element approximation of the solution of a population balance equation, which depends both the physical and internal property coordinates. We employ the operator-splitting method to split the high-dimensional population balance equation into two low-dimensional equations, and discretize the low-dimensional equations separately. In particular, we discretize the physical and internal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG) finite elements, respectively. It is demonstrated that the variational form of the operator-split population balance equation is equivalent to the variational form of the standard equation up to a perturbation term of order $tau^2$ in the backward Euler scheme, where $tau$ is a time step. Further, the stability and error estimates have been derived for the heterogeneous finite element discretization scheme applied to the population balance equation. It is shown that a slightly more regularity, $i.e,$ the mixed partial derivatives of the solution has to be bounded, is necessary for the solution of the population balance equation with the operator-splitting finite element method. Numerical results are presented to demonstrate the accuracy of the numerical scheme.