Abstract
We consider preconditioned iterative solution methods to solve the algebraic systems of equations arising from finite element discretizations of multiphase flow problems, based on the phase-field model.
The aim is to solve coupled physics problems, where both diffusive and convective processes take place simultaneously in time and space. To model the above, a coupled system of partial differential equations has to be solved, consisting of the Cahn-Hilliard equation to describe the diffusive interface and the time-dependent Navier-Stokes equation, to follow the evolution of the convection field in time.
We focus on the construction and efficiency of preconditioned iterative solution methods for the linear systems, arising after conforming and non-conforming finite element discretizations of the Cahn-Hilliard equation in space and implicit discretization schemes in time. The non-linearity of the phase-separation process is treated by Newton’s method. The resulting matrices admit a two-by-two block structure, utilized by the preconditioning techniques, proposed in the current work. We discuss approximation estimates of the preconditioners and include numerical experiments to illustrate their behaviour.
The work is supported by VR via the grant Finite element preconditioners for algebraic problems as arising in modelling of multiphase microstructures, 2009-2011. We also thank SNIC for resources provided through UPPMAX under Project p2009040, and the developers of the HSL MI-20 AMG package for kindly providing us with the AMG-Matlab interface.
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Boyanova, P., Do-Quang, M., Neytcheva, M. (2012). Block-Preconditioners for Conforming and Non-conforming FEM Discretizations of the Cahn-Hilliard Equation. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2011. Lecture Notes in Computer Science, vol 7116. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29843-1_62
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DOI: https://doi.org/10.1007/978-3-642-29843-1_62
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