Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation

Murat Uzunca; Ayşe Sarıaydın-Filibelioǧlu

doi:10.3934/naco.2020025

Article Contents

2021, Volume 11, Issue 2: 269-281. Doi: 10.3934/naco.2020025

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Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation

Murat Uzunca^1, , and
Ayşe Sarıaydın-Filibelioǧlu^2,

1.
Department of Mathematics, Sinop University, Sinop, 57000, Turkey
2.
Turkish Scientific and Technological Research Council (TÜBİTAK), Ankara, 06100, Turkey

^* Corresponding author: Murat Uzunca
^* Corresponding author: Murat Uzunca

Received: December 31, 2019

Revised: December 31, 2019

Early access: March 2020

Published: June 2021

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Abstract

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.

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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

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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

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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

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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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