曲面上梯度流模型的高效数值方法研究

梯度流模型是物理、材料、生物等领域中通过构造某种能量泛函求其极小而等价地建立起来的偏微分方程，其动力学行为是影响相场结构、性能及其稳定性的决定性因素。本课题拟深入研究曲面上梯度流模型的算法设计、理论分析及其实际应用。空间逼近采用曲面有限元方法、径向基函数方法等，时间离散采用自适应时间步长方法等。首先，研究基于能量凸分裂的计算方法，确保离散系统的求解等价于一个离散凸泛函的极小值问题，得到关于能量的绝对稳定性以及唯一可解性。其次，利用离散泛函的凸性构造最速下降法的预条件处理器，并结合多重网格方法降低求解离散系统的复杂度。最后，研究局部稳定化方法、特征线方法、通量修正输运方法、自适应方法等建立曲面上梯度流模型保物理性质的快速算法，进而精细刻画其动力学行为的时空图样生成及其形态演化过程。

The gradient flow models are partial differential equations equivalently established from the minimizer of some energy functionals in physical, material and biological field. Their dynamic behavior is the decisive factor affecting the phase field structure, performance and stability. This project aims to study the algorithm design, theoretical analysis and practical application of the gradient flow model on surfaces. Surface finite element method and radial basis function method are used for spatial approximation, respectively. And adaptive time step method is used for time discrete. First, we will focus on the numerical schemes based on energy functional convex splitting. The numerical solution of the discrete system is equivalent to the minimizer of some discrete convex functional, which ensures the unconditionally energy stability and unique solvability. Secondly, in order to reduce the complexity of solving discrete systems, multi-grid methods are combined with the preconditioner of the steepest descent method that is constructed by using the convexity of discrete functional. Finally, the surface gradient flow model with convection terms is studied. And the fast algorithms preserving physical properties are constructed by using the local stabilization method, the characteristic line method, the flux modified transport method and the adaptive method. Then spatial-temporal pattern formation and its morphology evolution of polymeric complex fluids can be accurately simulated.