相场方程的弱超内罚间断Galerkin方法及其自适应算法

Phase field equations are widely used in many areas, such as fluid dynamics, materials sciences, image processing and biomechanics. Allen-Cahn and Cahn-Hilliard equations are two important phase field modles. The main objective of this project is to study a weakly over-penalized interior penalty discontinuous Galerkin method for solving Allen-Cahn and Cahn-Hilliard equations. Compared with many well-known discontinuous Galerkin methods, the weakly over-penalized method is easy to implement and has less computational complexity. In addition, it has high intrinsic parallelism. First, we will obtain the a priori error estimates, which depend on the reciprocal of the perturbation parameter only in some lower polynomial order. Furthermore, in order to reduce the amount of calculation, we will study the a posteriori error estimates, and prove the upper and lower bounds of the error estimators. Finally, we shall design adaptive finite element methods according to the a posteriori error estimator to carry out numerical simulation.

相场方程在流体力学、材料科学、图像处理、生物力学等领域有着广泛应用。相场方程的两类重要模型问题是Allen-Cahn和Cahn-Hilliard方程。本课题拟研究求解Allen-Cahn和Cahn-Hilliard方程的一类弱超内罚间断Galerkin方法。与标准的内罚间断Galerkin方法相比，该方法更加容易实现，具有较少的计算复杂性，并且具有并行性的本质。项目首先研究先验误差，得到的估计只与摄动因子倒数的低阶多项式相关。为了有效减小计算量，项目将进一步研究弱超内罚间断Galerkin方法的后验误差，理论上证明后验误差估计子的上界和下界，并根据后验误差估计子设计自适应有限元方法进行数值模拟。

相场方程在流体力学、材料科学、图像处理、生物力学等领域有着广泛应用。相场方程的两类重要模型问题是Allen-Cahn和Cahn-Hilliard方程。项目首先构造了一类弱超内罚间断Galerkin方法求解Allen-Cahn方程，研究了该数值方法的最优先验误差估计，且所得估计只依赖于摄动因子倒数的低阶多项式。并推导了一类残量型后验误差估计，从理论上严格证明了后验误差估计子的上界，其下界可以通过数值实验验证。由于Allen-Cahn方程和线性抛物方程紧密相关，项目还研究了求解抛物方程的弱超内罚间断Galerkin方法的先验和后验误差估计。对Cahn-Hilliard方程的数值方法，由于方程是四阶问题，其误差分析有一定的困难，目前项目正在研究如何处理这些难点。另外，我们还研究了一类修正的弱Galerkin方法求解第一类变分不等式，获得了最优的先验误差估计。