Numerical solution of multi-variable cell population balance models. III. Finite element methods

We have developed finite element approximations to the generalized multi-variable cell population balance problem under conditions of changing substrate concentration. We considered different sets of basis functions consisting of 1st-, 2nd-, 3rd-, 4th-, 5th- and 6th-order polynomials. For the time integration of the nonlinear system of ordinary differential equations that result from the finite element discretization in space, we implemented both time-implicit and time-explicit algorithms. The set of basis functions consisting of 4th-order polynomials was found to be the most appropriate for all test problems considered. It offered more than a 10-fold reduction in required CPU time, when compared to the linear basis functions. Time-explicit methods were found to be preferable to time-implicit methods in terms of computational efficiency. Despite the fact that the required CPU time of the best finite element algorithm for single-variable simulations is comparable to that of the best spectral and finite difference method, the comparison is highly unfavorable for the finite element method in the case of multi-variable simulations.