Efficient Solvers of Discontinuous Galerkin Discretization for the Cahn–Hilliard Equations

In this paper, we develop and analyze a fast solver for the system of algebraic equations arising from the local discontinuous Galerkin (LDG) discretization and implicit time marching methods to the Cahn–Hilliard (CH) equations with constant and degenerate mobility. Explicit time marching methods fo...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 58; no. 2; pp. 380 - 408
Main Authors: Guo, Ruihan, Xu, Yan
Format: Journal Article
Language:English
Published: Boston Springer US 01.02.2014
Springer Nature B.V
Subjects:
Algebra
Algorithms
Approximation
Complexity
Computational Mathematics and Numerical Analysis
Discretization
Efficiency
Energy
Free energy
Galerkin method
Implicit methods
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Methods
Numerical methods
Ordinary differential equations
Partial differential equations
Solvers
Theoretical
Time marching
Local mode analysis
Convex splitting method
Diagonally implicit Runge–Kutta
65M60
Multigrid algorithm
FAS multigrid
Cahn–Hilliard equation
35K55
Local discontinuous Galerkin method
Additive Runge–Kutta
ISSN:0885-7474, 1573-7691
Online Access:Get full text
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Summary:In this paper, we develop and analyze a fast solver for the system of algebraic equations arising from the local discontinuous Galerkin (LDG) discretization and implicit time marching methods to the Cahn–Hilliard (CH) equations with constant and degenerate mobility. Explicit time marching methods for the CH equation will require severe time step restriction ( Δ t ∼ O ( Δ x 4 ) ) , so implicit methods are used to remove time step restriction. Implicit methods will result in large system of algebraic equations and a fast solver is essential. The multigrid (MG) method is used to solve the algebraic equations efficiently. The Local Mode Analysis method is used to analyze the convergence behavior of the linear MG method. The discrete energy stability for the CH equations with a special homogeneous free energy density Ψ ( u ) = 1 4 ( 1 − u 2 ) 2 is proved based on the convex splitting method. We show that the number of iterations is independent of the problem size. Numerical results for one-dimensional, two-dimensional and three-dimensional cases are given to illustrate the efficiency of the methods. We numerically show the optimal complexity of the MG solver for P 1 element. For P 2 approximation, the optimal or sub-optimal complexity of the MG solver are numerically shown.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-013-9738-4