SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Structure Preserving Discontinuous Galerkin methods in space and time for Allen-Cahn equation

A. Sariaydin, B. Karasözen and M. Uzunca

Abstract
Gradient flow equations like Allen Cahn and Cahn-Hilliard are characterized by the energy decreasing property. We apply space-time discontinuous finite elements to preserve the energy decreasing property in the discretized form. For discretization in space, we use the interior penalty method discontinuous Galerkin method [1]. The semi-discretized system is solved by the discontinuous Galerkin-Petrov method [2], which is preserving the energy decreasing property and is A-stable. Numerical results for Allen-Cahn equation with Neumannn and periodic boundary conditions confirm the theoretical results.

Bibliography
[1] D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), pp. 1749-1779.
[2] F. Schieweck, A-stable discontinuous Galerkin-Petrov time discretization of higher order, Journal of Numerical Mathematics, 18 (2010), pp. 25-57.

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