Finite element approximation of nematic liquid crystal flows using a saddle-point structure

Badia, Santiago; Guillén-González, Francisco; Gutiérrez-Santacreu, Juan Vicente

doi:10.1016/j.jcp.2010.11.033

Cited by 79 publications

(55 citation statements)

References 30 publications

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“…This idea has been applied to our target problem, (thermally coupled) inductionless MHD problem, (where the unknowns are the velocity, pressure, current density and electric potential) but can also be applied to other problems like resistive MHD [4] or liquid crystal problems [5]. We consider different preconditioners based on approximations of the resulting Schur complement matrices.…”

Section: Discussionmentioning

confidence: 99%

Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem

Badia

Martı́n

Planas

2014

Journal of Computational Physics

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The thermally coupled incompressible inductionless magnetohydrodynamics (MHD) problem models the flow of an electrically charged fluid under the influence of an external electromagnetic field with thermal coupling. This system of partial differential equations is strongly coupled and highly nonlinear for real cases of interest. Therefore, fully implicit time integration schemes are very desirable in order to capture the different physical scales of the problem at hand. However, solving the multiphysics linear systems of equations resulting from such algorithms is a very challenging task which requires efficient and scalable preconditioners. In this work, a new family of recursive block LU preconditioners is designed and tested for solving the thermally coupled inductionless MHD equations. These preconditioners are obtained after splitting the fully coupled matrix into one-physics problems for every variable (velocity, pressure, current density, electric potential and temperature) that can be optimally solved, e.g., using preconditioned domain decomposition algorithms. The main idea is to arrange the original matrix into an (arbitrary) 2 × 2 block matrix, and consider a LU preconditioner obtained by approximating the corresponding Schur complement. For every one of the diagonal blocks in the LU preconditioner, if it involves more than one type of unknown, we proceed the same way in a recursive fashion. This approach is stated in an abstract way, and can be straightforwardly applied to other multiphysics problems. Further, we precisely explain a flexible and general software design for the code implementation of this type of preconditioners.

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Section: Discussionmentioning

confidence: 99%

Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem

Badia

Martı́n

Planas

2014

Journal of Computational Physics

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“…In [6], a Lagrange multiplier is introduced to penalize the restriction of unitary director vector in the Nematic Liquid Crystals framework and this idea was extended to the Cahn-Hilliard equation in [37,38] to derive unconditionally energy-stable (for a modified energy) linear schemes. Indeed, a one-step and first order in time scheme and two two-step and second order schemes were introduced.…”

Section: Second Order Lagrange Multiplier Scheme (Lm2)mentioning

confidence: 99%

“…In [6] the authors introduced r = (|d| 2 − 1)/ε 2 as an approximation of the Lagrange multiplier associated to the restriction |d| = 1, and then the Ginzburg-Landau function can be rewritten as…”

Section: First Order Lagrange Multiplier Scheme (Lm1)mentioning

confidence: 99%

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Numerical Methods for Solving the Cahn–Hilliard Equation and Its Applicability to Related Energy-Based Models

Tierra

Guillén-González

2014

Arch Computat Methods Eng

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In this paper, we review some numerical methods presented in the literature in the last years to approximate the Cahn-Hilliard equation. Our aim is to compare the main properties of each one of the approaches to try to determine which one we should choose depending on which are the crucial aspects when we approximate the equations. Among the properties that we consider desirable to control are the time accuracy order, energy-stability, unique solvability and the linearity or nonlinearity of the resulting systems. In particular, we concern about the iterative methods used to approximate the nonlinear schemes and the constraints that may arise on the physical and computational parameters.Furthermore, we present the connections of the Cahn-Hilliard equation with other physically motivated systems (not only phase field models) and we state how the ideas of efficient numerical schemes in one topic could be extended to other frameworks in a natural way.

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“…In order to get numerical methods with an associated energy law, the sphere constraint is usually penalized with the Ginzburg-Landau penalty function such as in the works of Becker, Feng and Prohl [8], and Lin, Liu, and Zhang [45]. An alternative recently proposed by Badia, Guillén-González and Gutiérrez-Santacreu in [5] is to use an equivalent saddle-point formulation of the system proposed by Lin in [41]. It provides a system of partial differential equations equivalent to the one in [41] that also leads to numerical methods with an energy law.…”

Section: Introductionmentioning

confidence: 99%

An Overview on Numerical Analyses of Nematic Liquid Crystal Flows

Badia

Guillén-González

Gutiérrez-Santacreu

2011

Arch Computat Methods Eng

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The purpose of this work is to provide an overview of the most recent numerical developments in the field of nematic liquid crystals. The Ericksen-Leslie equations govern the motion of a nematic liquid crystal. This system, in its simplest form, consists of the NavierStokes equations coupled with an extra anisotropic stress tensor, which represents the effect of the nematic liquid crystal on the fluid, and a convective harmonic map equation. The sphere constraint must be enforced almost everywhere in order to obtain an energy estimate.Since an almost everywhere satisfaction of this restriction is not appropriate at a numerical level, two alternative approaches have been introduced: a penalty method and a saddle-point method. These approaches are suitable for their numerical approximation by finite elements, since a discrete version of the restriction is enough to prove the desired energy estimate.The Ginzburg-Landau penalty function is usually used to enforce the sphere constraint.Finite element methods of mixed type will play an important role when designing numerical approximations for the penalty method in order to preserve the intrinsic energy estimate.The inf-sup condition that makes the saddle-point method well-posed is not clear yet.The only inf-sup condition for the Lagrange multiplier is obtained in the dual space of H 1 (Ω).But such an inf-sup condition requires more regularity for the director vector than the one provided by the energy estimate. Herein, we will present an alternative inf-sup condition whose proof for its discrete counterpart with finite elements is still open.2000 Mathematics Subject Classification. 35Q35, 65M12, 65M60.

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Finite element approximation of nematic liquid crystal flows using a saddle-point structure

Cited by 79 publications

References 30 publications

Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem

Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem

Numerical Methods for Solving the Cahn–Hilliard Equation and Its Applicability to Related Energy-Based Models

An Overview on Numerical Analyses of Nematic Liquid Crystal Flows

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