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

• Volume: 46, Issue: 6, page 1447-1465
• ISSN: 0764-583X

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## Abstract

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We present a heterogeneous finite element method for the solution of a high-dimensional population balance equation, which depends both the physical and the internal property coordinates. The proposed scheme tackles the two main difficulties in the finite element solution of population balance equation: (i) spatial discretization with the standard finite elements, when the dimension of the equation is more than three, (ii) spurious oscillations in the solution induced by standard Galerkin approximation due to pure advection in the internal property coordinates. The key idea is to split the high-dimensional population balance equation into two low-dimensional equations, and discretize the low-dimensional equations separately. In the proposed splitting scheme, the shape of the physical domain can be arbitrary, and different discretizations can be applied to the low-dimensional equations. In particular, we discretize the physical and internal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG) finite elements, respectively. The stability and error estimates of the Galerkin/SUPG finite element discretization of the population balance equation are derived. 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 optimal order of convergence. Numerical results are presented to support the analysis.

## How to cite

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Ganesan, Sashikumaar. "An operator-splitting Galerkin/SUPG finite element method for population balance equations : stability and convergence." ESAIM: Mathematical Modelling and Numerical Analysis 46.6 (2012): 1447-1465. <http://eudml.org/doc/222160>.

@article{Ganesan2012,
abstract = {We present a heterogeneous finite element method for the solution of a high-dimensional population balance equation, which depends both the physical and the internal property coordinates. The proposed scheme tackles the two main difficulties in the finite element solution of population balance equation: (i) spatial discretization with the standard finite elements, when the dimension of the equation is more than three, (ii) spurious oscillations in the solution induced by standard Galerkin approximation due to pure advection in the internal property coordinates. The key idea is to split the high-dimensional population balance equation into two low-dimensional equations, and discretize the low-dimensional equations separately. In the proposed splitting scheme, the shape of the physical domain can be arbitrary, and different discretizations can be applied to the low-dimensional equations. In particular, we discretize the physical and internal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG) finite elements, respectively. The stability and error estimates of the Galerkin/SUPG finite element discretization of the population balance equation are derived. 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 optimal order of convergence. Numerical results are presented to support the analysis.},
author = {Ganesan, Sashikumaar},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Population balance equations; operator-splitting method; error analysis; streamline Upwind Petrov Galerkin finite element methods; backward Euler scheme; convergence; population balance equation; streamline-upwind Petrov Galerkin finite element method; stability; accuracy; diffusion-convection equation; consistency; error estimate; numerical results},
language = {eng},
month = {5},
number = {6},
pages = {1447-1465},
publisher = {EDP Sciences},
title = {An operator-splitting Galerkin/SUPG finite element method for population balance equations : stability and convergence},
url = {http://eudml.org/doc/222160},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Ganesan, Sashikumaar
TI - An operator-splitting Galerkin/SUPG finite element method for population balance equations : stability and convergence
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/5//
PB - EDP Sciences
VL - 46
IS - 6
SP - 1447
EP - 1465
AB - We present a heterogeneous finite element method for the solution of a high-dimensional population balance equation, which depends both the physical and the internal property coordinates. The proposed scheme tackles the two main difficulties in the finite element solution of population balance equation: (i) spatial discretization with the standard finite elements, when the dimension of the equation is more than three, (ii) spurious oscillations in the solution induced by standard Galerkin approximation due to pure advection in the internal property coordinates. The key idea is to split the high-dimensional population balance equation into two low-dimensional equations, and discretize the low-dimensional equations separately. In the proposed splitting scheme, the shape of the physical domain can be arbitrary, and different discretizations can be applied to the low-dimensional equations. In particular, we discretize the physical and internal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG) finite elements, respectively. The stability and error estimates of the Galerkin/SUPG finite element discretization of the population balance equation are derived. 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 optimal order of convergence. Numerical results are presented to support the analysis.
LA - eng
KW - Population balance equations; operator-splitting method; error analysis; streamline Upwind Petrov Galerkin finite element methods; backward Euler scheme; convergence; population balance equation; streamline-upwind Petrov Galerkin finite element method; stability; accuracy; diffusion-convection equation; consistency; error estimate; numerical results
UR - http://eudml.org/doc/222160
ER -

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