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

Published online by Cambridge University Press:  31 May 2012

Sashikumaar Ganesan
Sashikumaar Ganesan*
Affiliation:
Numerical Mathematics and Scientific Computing, Supercomputer Education and Research Centre, Indian Institute of Science, 560012 Bangalore, India. sashi@serc.iisc.ac.in
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Abstract

We present a heterogeneous finite element method for the solution of a high-dimensionalpopulation balance equation, which depends both the physical and the internal propertycoordinates. The proposed scheme tackles the two main difficulties in the finite elementsolution of population balance equation: (i) spatial discretization with the standardfinite elements, when the dimension of the equation is more than three, (ii) spuriousoscillations in the solution induced by standard Galerkin approximation due to pureadvection in the internal property coordinates. The key idea is to split thehigh-dimensional population balance equation into two low-dimensional equations, anddiscretize the low-dimensional equations separately. In the proposed splitting scheme, theshape of the physical domain can be arbitrary, and different discretizations can beapplied to the low-dimensional equations. In particular, we discretize the physical andinternal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG)finite elements, respectively. The stability and error estimates of the Galerkin/SUPGfinite element discretization of the population balance equation are derived. It is shownthat a slightly more regularity, i.e.the mixed partial derivatives of the solution has to be bounded, is necessary for theoptimal order of convergence. Numerical results are presented to support the analysis.

Keywords

Population balance equationsoperator-splitting methoderror analysisstreamline Upwind Petrov Galerkin finite element methodsbackward Euler scheme
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 6 , November 2012 , pp. 1447 - 1465
Copyright
© EDP Sciences, SMAI, 2012

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