A novel free–energy stable discontinuous Galerkin method is developed for the Cahn–Hilliard equation with non–conforming elements. This work focuses on dynamic polynomial adaptation (p–refinement) and constitutes an extension of the method developed by Manzanero et al. in Journal of Computational Physics 403:109072, 2020, which makes use of the summation–by–parts simultaneous–approximation term technique along with Gauss–Lobatto points and the Bassi–Rebay 1 (BR1) scheme. The BR1 numerical flux accommodates non–conforming elements, which are connected through the mortar method. The scheme has been analytically proven to retain its free–energy stability when transitioning to non-conforming elements. Furthermore, a methodology to perform the adaptation is introduced based on the knowledge of the location of the interface between phases. The adaptation methodology is tested for its accuracy and effectiveness through a series of steady and unsteady test cases. We test the scheme for freestream preservation and primary quantity conservation on non–conforming curvilinear meshes. We solve a steady one–dimensional interface test case to initially examine the accuracy of the adaptation. Furthermore, we study the formation of a static bubble in two dimensions and verify that the accuracy of the solver is maintained while the degrees of freedom decrease to less than half compared to the uniform solution. Lastly, we examine an unsteady case such as the spinodal decomposition and show that the same results for the free–energy are recovered, with a 35% reduction of the degrees of freedom for the two–dimensional case considered and a 48% reduction for the three–dimensional case.

针对具有不合格元素的Cahn-Hilliard方程，开发了一种新颖的自由能稳定不连续Galerkin方法。这项工作着重于动态多项式自适应（p细化），并且构成了Manzanero等人开发的方法的扩展。见《计算物理杂志》 403：109072，2020年，该书利用了分部加总的同时逼近项技术以及高斯-洛巴托点和Bassi-Rebay 1（BR1）方案。BR1数值通量可容纳不合格元素，这些元素通过砂浆法连接。经分析证明，该方案在过渡到不合格元素时仍可保持其自由能稳定性。此外，基于相之间的界面的位置的知识，引入了执行自适应的方法。通过一系列稳定和不稳定的测试案例，对适应性方法的准确性和有效性进行了测试。我们在不合格的曲线网格上测试了自由流保留和一次数量守恒的方案。我们解决了一个稳定的一维界面测试用例，以初步检查自适应的准确性。此外，我们研究了二维静态气泡的形成，并验证了求解器的精度得以保持，而与统一解决方案相比，其自由度减小到不到一半。最后，我们研究了一个不稳定的情况，例如旋节线分解，结果表明，对于自由能，可以得到相同的结果，对于二维情况，自由度降低了35％，对于二维情况，自由度降低了48％。三维情况。