# 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 -

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