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

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

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

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topGanesan, 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 -

## References

top- S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 3th edition. Springer (2008). Zbl1135.65042
- E. Burman, Consistent SUPG-method for transient transport problems : Stability and convergence. Comput. Methods Appl. Mech. Eng.199 (2010) 1114–1123. Zbl1227.76047
- I.T. Cameron, F.Y. Wang, C.D. Immanuel and F. Stepanek, Process systems modelling and applications in granulation : a review. Chem. Eng. Sci.60 (2005) 3723–375.
- F.B. Campos and P.L.C. Lage, A numerical method for solving the transient multidimensional population balance equation using an Euler-Lagrange formulation. Chem. Eng. Sci.58 (2003) 2725–2744.
- P. Chen, J. Sanyal and M.P. Dudukovic, CFD modeling of bubble columns flows : implementation of population balance. Chem. Eng. Sci.59 (2004) 5201–5207.
- K. Eriksson and C. Johnson, Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems. Math. Comput.60 (1993) 167–188. Zbl0795.65074
- S. Ganesan and L. Tobiska, Implementation of an operator-splitting finite element method for high-dimensional parabolic problems. Faculty of Mathematics, University of Magdeburg, Preprint No. 11-04 (2011).
- S. Ganesan and L. Tobiska, An operator-splitting finite element method for the efficient parallel solution of multidimensional population balance systems. Chem. Eng. Sci.69 (2012) 59–68.
- R. Glowinski, E.J. Dean, G. Guidoboni, D.H. Peaceman and H.H. Rachford, Applications of operator-splitting methods to the direct numerical simulation of particulate and free-surface flows and to the numerical solution of the two-dimensional elliptic Monge-Ampère equation. Japan J. Ind. Appl. Math.25 (2008) 1–63.
- R. Gunawan, I. Fusman and R.D. Braatz, High resolution algorithms for multidimensional population balance equations. AIChE J.50 (2004) 2738–2749.
- R. Gunawan, I. Fusman and R.D. Braatz, Parallel high-resolution finite volume simulation of particulate processes. AIChE J.54 (2008) 1449–1458.
- T.J.R. Hughes and A.N. Brooks, A multi-dimensional upwind scheme with no cross-wind diffusion, in Finite element methods for convection dominated flows, edited by T.J.R. Hughes. ASME, New York (1979) 19–35.
- H.M. Hulburt and S. Katz, Some problems in particle technology : A statistical mechanical formulation. Chem. Eng. Sci.19 (1964) 555–574.
- V. John and J. Novo, Error analysis of the SUPG finite element discretization of evolutionary convection-diffusion-reaction equations. SIAM J. Numer. Anal.49 (2011) 1149–1176. Zbl1233.65065
- V. John, M. Roland, T. Mitkova, K. Sundmacher, L. Tobiska and A. Voigt, Simulations of population balance systems with one internal coordinate using finite element methods. Chem. Eng. Sci.64 (2009) 733–741.
- V. Kulikov, H. Briesen, R. Grosch, A. Yang, L. von Wedel and W. Marquardt, Modular dynamic simulation for integrated particulate processes by means of tool integration. Chem. Eng. Sci.60 (2005) 2069–2083.
- V. Kulikov, H. Briesen and W. Marquardt, A framework for the simulation of mass crystallization considering the effect of fluid dynamics. Chem. Eng. Sci.45 (2006) 886–899.
- G. Lian, S. Moore and L. Heeney, Population balance and computational fluid dynamics modelling of ice crystallisation in a scraped surface freezer. Chem. Eng. Sci.61 (2006) 7819–7826.
- D.L. Ma, D.K. Tafti and R.D. Braatz, High-resolution simulation of multidimensional crystal growth. Ind. Eng. Chem. Res.41 (2002) 6217–6223.
- D.L. Ma, D.K. Tafti and R.D. Braatz, Optimal control and simulation of multidimensional crystallization processes. Comput. Chem. Eng.26 (2002) 1103–1116.
- A. Majumder, V. Kariwala, S. Ansumali and A. Rajendran, Fast high-resolution method for solving multidimensional population balances in crystallization. Ind. Eng. Chem. Res.49 (2010) 3862–3872.
- D. Marchisio and R. Fox, Solution of population balance equations using the direct quadrature method of moments. J. Aero. Sci.36 (2005) 43–73.
- D. L. Marchisio and R.O. Fox, Solution of population balance equations using the direct quadrature method of moments. J. Aero. Sci.36 (2005) 43–73.
- M. N. Nandanwara and S. Kumar, A new discretization of space for the solution of multi-dimensional population balance equations : Simultaneous breakup and aggregation of particles. Chem. Eng. Sci.63 (2008) 3988–3997.
- D. Ramkrishna, Population Balances, Theory and Applications to Particulate Systems in Engineering. Academic Press, San Diego (2000).
- D. Ramkrishna and A.W. Mahoney, Population balance modeling : Promise for the future. Chem. Eng. Sci.57 (2002) 595–606.
- V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, 3th edition. Springer (2008). Zbl1105.65102

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