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.

中文翻译：

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平流 Allen-Cahn 方程的自适应不连续 Galerkin 有限元

我们应用空间自适应内罚不连续伽辽金方法来求解具有膨胀和收缩速度场的平流 Allen-Cahn 方程。平流 Allen-Cahn 方程首先在时间上离散，并且使用基于残差的后验误差估计器通过自适应算法求解得到的半线性椭圆偏微分方程。由于非发散自由速度场，后验误差估计器包含附加项。数值例子通过准确地解析锐利层来证明自适应方法的有效性和准确性。