### 2011-05

by Ahmed, N.; Matthies, G.; Tobiska, L..

**Series:** 2011-05, Preprints

- MSC:
- 65M12 Stability and convergence of numerical methods
- 65M60 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods
- 65M15 Error bounds

**Abstract:**

In population balance equations, the distribution of the entities depends not only on space and time but also on their own properties referred to as internal coordinates. The operator splitting method is used to transform the whole time-dependent problem into two unsteady subproblems of a smaller complexity. The first subproblem is a time-dependent convection-diffusion problem while the second one is a transient transport problem with pure advection. We use the backward Euler method to discretize the subproblems in time. Since the first problem is convection-dominated, the local projection method is applied as stabilization in space. The transport problem in the one-dimensional internal coordinate is solved by a discontinuous Galerkin method. The unconditional stability of the method will be presented. Optimal error estimates are given. Numerical results confirm the theoretical predictions.

**Keywords:**

Operator splitting; discontinuous Galerkin; stabilized finite elements; population balance equations