A Preconditioner for the Ohta--Kawasaki Equation

Farrell, Patrick E.; Pearson, John W.

doi:10.1137/16m1065483

Cited by 13 publications

(14 citation statements)

References 17 publications

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“…32,33 In terms of numerical strategies explicitly designed for the NCH equation, the literature is not as rich as the case of the CH equation. We highlight the development of preconditioners, 30,38,46 computational implementation details, and theoretical proofs regarding stability, boundedness, and mass conservation using the finite element method. 36 Several strategies have been proposed to increase the accuracy and performance of the simulations.…”

Section: Governing Equationsmentioning

confidence: 99%

“…Thus standard C 0 -continuous finite elements are not suitable for its primal variational formulation. Nevertheless, a splitting strategy, also called mixed formulation, is employed to avoid the continuity constraint and enable the use of C 0 -continuous elements to approximate the solution of the CH 43 and the NCH 30,36 equations, converting the nonlinear equation into a coupled nonlinear system, with two degrees of freedom per node.…”

Section: Spatial Discretizationmentioning

confidence: 99%

“…5,[26][27][28][29] However, the use of these numerical simulations is nontrivial and can present difficulties related to the stiffness of the equations and demand large computational power. 30 In this sense, the development of computational techniques that improve the performance and accuracy of the computational modeling of the NCH equation is an active topic of research.…”

Section: Introductionmentioning

confidence: 99%

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Finite element solution of nonlocal Cahn–Hilliard equations with feedback control time step size adaptivity

Barros

Cortês

Coutinho

2021

Numerical Meth Engineering

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In this study, we evaluate the performance of feedback control-based time step adaptivity schemes for the nonlocal Cahn-Hilliard equation derived from the Ohta-Kawasaki free energy functional. The temporal adaptivity scheme is recast under the linear feedback control theory equipped with an error estimation that extrapolates the solution obtained from an energy-stable, fully implicit time marching scheme. We test three time step controllers with different properties: a simple Integral controller, a complete proportional-integral-derivative controller, and the PC11 predictive controller. We assess the performance of the adaptive schemes for the nonlocal Cahn-Hilliard equation in terms of the number of time steps required for the complete simulation and the computational effort measured by the required number of nonlinear and linear solver iterations. We also present numerical evidence of mass conservation and free energy decay for simulations with the three different time step controllers. The PC11 predictive controller is the best in all three-dimensional test cases.

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Section: Governing Equationsmentioning

confidence: 99%

Section: Spatial Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Finite element solution of nonlocal Cahn–Hilliard equations with feedback control time step size adaptivity

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Cortês

Coutinho

2021

Numerical Meth Engineering

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“…In fact, we need to do numerical calculations on domains with at least a few dozen periodic structures, such as crystal nucleation and growth. In order to improve computational efficiency, some researchers have worked hard to present various preconditioners [47,[54][55][56][57][58]. To reduce the computation burden, effective utilization of adaptive mesh refinement to PFC model has been shown [48].…”

Section: Adaptive Fem To Pfc Modelmentioning

confidence: 99%

Error analysis of SAV finite element method to phase field crystal model

Wang,

Huang,

Jiang

2019

Preprint

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In this paper, we construct and analyze an energy stable scheme by combining the latest developed scalar auxiliary variable (SAV) approach and linear finite element method (FEM) for phase field crystal (PFC) model, and show rigorously that the scheme is first-order in time and second-order in space for the L 2 and H −1 gradient flow equations. To reduce efficiently computational cost and capture accurately the phase interface, we give a simple adaptive strategy, equipped with a posteriori gradient estimator, i.e. L 2 norm of the recovered gradient. Extensive numerical experiments are presented to verify our theoretical results and to demonstrate the effectiveness and accuracy of our proposed method.

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“…Many of the preconditioners for block systems based on block factorisations require discretisations of differential operators not contained in the original problem. These include the pressure-convection-diffusion (PCD) approximation for Navier-Stokes [25,16], and preconditioners for models of phase separation [19,24]. An alternate approach to derive preconditioners for block systems is to use arguments from functional analysis to arrive at block-diagonal preconditioners.…”

mentioning

confidence: 99%

Solver Composition Across the PDE/Linear Algebra Barrier

Kirby¹,

Mitchell²

2018

SIAM J. Sci. Comput.

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Abstract. The efficient solution of discretizations of coupled systems of partial differential equations (PDEs) is at the core of much of numerical simulation. Significant effort has been expended on scalable algorithms to precondition Krylov iterations for the linear systems that arise. With few exceptions, the reported numerical implementation of such solution strategies is specific to a particular model setup, and intimately ties the solver strategy to the discretization and PDE, especially when the preconditioner requires auxiliary operators. In this paper, we present recent improvements in the Firedrake finite element library that allow for straightforward development of the building blocks of extensible, composable preconditioners that decouple the solver from the model formulation. Our implementation extends the algebraic composability of linear solvers offered by the PETSc library by augmenting operators, and hence preconditioners, with the ability to provide any necessary auxiliary operators. Rather than specifying up front the full solver configuration tied to the model, solvers can be developed independently of model formulation and configured at runtime. We illustrate with examples from incompressible fluids and temperature-driven convection.Key words. iterative methods, preconditioning, composable solvers, multiphysics AMS subject classifications. 65N22, 65F08, 65F10 DOI. 10.1137/17M11332081. Introduction. For over a decade, domain-specific languages for numerical partial differential equations (henceforth PDEs) such as Sundance [30,29], FEniCS [28], and now Firedrake [40] have enabled users to efficiently generate algebraic systems from a high-level description of the variational problems. Both FEniCS and Firedrake make use of the Unified Form Language (UFL) [1] as a description language for the weak forms of PDEs, converting it into efficient low-level code for form evaluation. They also share a Python interface that, for the intersection of their feature sets, is nearly source-compatible. These high-level PDE codes succeed by connecting a rich description language for PDEs to effective lower-level solver libraries such as PETSc [4,5] or Trilinos [21] for the requisite, and performance-critical, numerical (non)linear algebra.These high-level PDE projects utilize the solver packages in an essentially unidirectional way: the residuals are evaluated, Jacobians formed, and are then handed off to mainly algebraic techniques. Hence, the codes work at their best when (compositions of) existing black-box matrix techniques effectively solve the algebraic systems. However, in many situations the best preconditioners require additional structure

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A Preconditioner for the Ohta--Kawasaki Equation

Cited by 13 publications

References 17 publications

Finite element solution of nonlocal Cahn–Hilliard equations with feedback control time step size adaptivity

Finite element solution of nonlocal Cahn–Hilliard equations with feedback control time step size adaptivity

Error analysis of SAV finite element method to phase field crystal model

Solver Composition Across the PDE/Linear Algebra Barrier

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