A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver

1.
Statistical and Applied Mathematical Sciences Institute (SAMSI), Research Triangle Park, NC 27709
2.
Department of Mathematics, University of Tennessee, Knoxville, TN 37996
3.
Department of Mathematics, The University of Tennessee, Knoxville, TN 37996-0612

Received: June 2013

Revised: August 2013

Published: November 2013

Abstract/ Introduction Related Papers Cited by

Abstract

Abstract
In this paper we devise and analyze a mixed discontinuous Galerkin finite element method for a modified Cahn-Hilliard equation that models phase separation in diblock copolymer melts. The time discretization is based on a convex splitting of the energy of the equation. We prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system, unconditionally uniquely solvable, and convergent in the natural energy norm with optimal rates. We describe an efficient nonlinear multigrid solver for advancing our semi-implicit scheme in time and conclude the paper with numerical tests confirming the predicted convergence rates and suggesting the near-optimal complexity of the solver.
- Cahn-Hilliard equation,
- diblock copolymer,
- discontinuous Galerkin,
- symmetric interior penalty,
- convex splitting,
- energy stability,
- convergence,
- nonlinear multigrid.
Mathematics Subject Classification:Primary: 65M60, 65M12; Secondary: 65M55.
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