Abstract

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 $p\geq1$ and backward Euler discretization in time, we show that the order-parameter c is approximated in the broken ${\rm L}^\infty({\rm H}^1)$ norm, with optimal order ${\cal O}(h^p+\tau)$; the associated chemical potential $w=\Phi'(c)-\gamma^2\Delta c$ is shown to be approximated, with optimal order ${\cal O}(h^p+\tau)$ in the broken ${\rm L}^2({\rm H}^1)$ norm. Here $\Phi(c)=\frac{1}{4}(1-c^2)^2$ is a quartic free-energy function and $\gamma>0$ is an interface parameter. Numerical results are presented with polynomials of degree $p=1,2,3$.

MSC codes

  1. 65M12
  2. 65M15
  3. 65M60

Keywords

  1. Cahn–Hilliard equation
  2. discontinuous Galerkin finite element method
  3. convergence
  4. error analysis

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Information & Authors

Information

Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 2660 - 2685
ISSN (online): 1095-7170

History

Submitted: 10 June 2008
Accepted: 26 February 2009
Published online: 31 July 2009

MSC codes

  1. 65M12
  2. 65M15
  3. 65M60

Keywords

  1. Cahn–Hilliard equation
  2. discontinuous Galerkin finite element method
  3. convergence
  4. error analysis

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