An operator-splitting Galerkin/SUPG finite element method for population balance equations: stability and convergence.

*(English)*Zbl 1273.65143The author considers a population balance equation, i.e., a diffusion-convection equation (with constant diffusion coefficient \(\epsilon\)) containing also the gradient with respect to additional unknowns other than the spatial coordinates, like length and particle size. So, the equation is posed in a high-dimensional space, along with first-kind boundary and initial data. It is assumed that the velocity is divergence-free and does not depend on the additional unknowns, and the growth rate vector (the factor of the additional gradient) is divergence-free in its unknowns. For this problem, a streamline-upwind Petrov-Galerkin (SUPG) approximation is proposed and the ellipticity of the corresponding bilinear form is shown. For the time discretization, the backward Euler method is applied, but to struggle against the high dimensionality of the resulting task, a simple time splitting is envisaged in which a step solving the transport equation in the additional coordinates using SUPG is followed by a step on the diffusion-convection equation using the standard Galerkin (assuming \(\epsilon\) to be large enough to cause no problems). This method is written in equivalent one-step form the consistency of which and the ellipticity of the corresponding bilinear form are proved. Assuming some more regularity than usual (in a norm containing also the mixed derivatives with respect to spatial and additional unknowns), an error estimate is derived for conforming finite elements (which should conserve the divergence-freeness of velocity and growth vector). A purely academic example shows good numerical results for \(Q_1-P_1\) and \(Q_2-P_2\) elements, 1 additional and 2 spatial unknowns.

Reviewer: Gisbert Stoyan (Budapest)

##### MSC:

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

35K20 | Initial-boundary value problems for second-order parabolic equations |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |