DISCONTINUOUS GALERKIN FINITE ELEMENT APPROXIMATION OF THE CAHN–HILLIARD EQUATION WITH CONVECTION

The paper is concerned with the construction and convergence analysis of a discontinuous Galerkin finite element method for the Cahn–Hilliard equation with convection. Using discontinuous piecewise polynomials of degree p ≥ 1 and backward Euler discretization in time, we show that the order-paramete...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 2009-01, Vol.47 (4), p.2660-2685
Main Authors: KAY, DAVID, STYLES, VANESSA, SÜLI, ENDRE
Format: Article
Language:English
Subjects:
Approximation
Convection
Error bounds
Estimation methods
Finite element method
Galerkin methods
Mathematical analysis
Mathematical discontinuity
Mathematical functions
Mathematical models
Methods
Norms
Optimization
Phase transitions
Polynomials
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Summary:The paper is concerned with the construction and convergence analysis of a discontinuous Galerkin finite element method for the Cahn–Hilliard equation with convection. Using discontinuous piecewise polynomials of degree p ≥ 1 and backward Euler discretization in time, we show that the order-parameter c is approximated in the broken L ∞ (H 1 ) norm, with optimal order O (h p + τ); the associated chemical potential ω = Φ′ (c) – γ 2 Δc is shown to be approximated, with optimal order O (h p + τ) in the broken L 2 (H 1 ) norm. Here Φ (c) = ¼ (1 – c 2 ) 2 is a quartic free-energy function and γ > 0 is an interface parameter. Numerical results are presented with polynomials of degree p = 1, 2, 3.
ISSN:0036-1429
1095-7170
DOI:10.1137/080726768