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Licensed Unlicensed Requires Authentication Published by De Gruyter December 8, 2015

High order approximations in space and time of a sixth order Cahn–Hilliard equation

  • Oleg Boyarkin , Ronald H. W. Hoppe EMAIL logo and Christopher Linsenmann

Abstract

We consider an initial-boundary value problem for a sixth order Cahn-Hilliard equation describing the formation of microemulsions. Based on a Ciarlet-Raviart type mixed formulation as a system consisting of a second order and a fourth order equation, the spatial discretization is done by a C0 Interior Penalty Discontinuous Galerkin (C0IPDG) approximation with respect to a geometrically conforming simplicial triangulation of the computational domain. The DG trial spaces are constructed by C0 conforming Lagrangian finite elements of polynomial degree k ≥ 2. This leads to an initial value problem for an index 1 Differential Algebraic Equation (DAE) which is further discretized in time by an s-stage Diagonally Implicit Runge-Kutta (DIRK) method of order p ≥ 2. The resulting parameter dependent nonlinear algebraic system is solved numerically by a predictor-corrector continuation strategy with constant continuation as a predictor and New- ton’s method as a corrector featuring an adaptive choice of the continuation parameter. Numerical results illustrate the performance of the suggested approach.

Received: 2015-9-25
Accepted: 2015-10-15
Published Online: 2015-12-8
Published in Print: 2015-12-1

© 2015 by Walter de Gruyter Berlin/Boston

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