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Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation

  • * Corresponding author: Murat Uzunca

    * Corresponding author: Murat Uzunca 
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  • We apply a space adaptive interior penalty discontinuous Galerkin method for solving advective Allen-Cahn equation with expanding and contracting velocity fields. The advective Allen-Cahn equation is first discretized in time and the resulting semi-linear elliptic PDE is solved by an adaptive algorithm using a residual-based a posteriori error estimator. The a posteriori error estimator contains additional terms due to the non-divergence-free velocity field. Numerical examples demonstrate the effectiveness and accuracy of the adaptive approach by resolving the sharp layers accurately.

    Mathematics Subject Classification: 65M20, 65M22, 65M50, 65M60.


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  • Figure 1.  Expanding flow: (Top) Uniform solutions, (middle) adaptive solutions and (bottom) adaptive meshes at time instances $ t = 0 $, $ t = 0.03 $ and $ t = 0.06 $ from left to right

    Figure 2.  Expanding flow: (Left) Maximum element error propagation over time, and (right) evaluation of DoFs over time; dashed line indicates the DoFs used for uniform solutions

    Figure 3.  Sheering flow: (Top) Uniform solutions, (middle) adaptive solutions and (bottom) adaptive meshes at time instances $ t = 0 $, $ t = 0.01 $ and $ t = 0.06 $ from left to right

    Figure 4.  Sheering flow: (Left) Maximum element error propagation over time, and (right) evaluation of DoFs over time; dashed line indicates the DoFs used for uniform solutions

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