自适应三角谱元方法及其应用

The goal of the project is to develop an adaptive triangular spectral element method for the simulation of incompressible fluids in complex geometry. Triangular spectral element method is different from the classical spectral element method by using triangular(tetrahedron) element instead of quadrilateral(hexahedron) element, which makes it more suitable for developing an adaptive mesh method. We will use the triangulation technique to generate an initial mesh of triangles, then a refining/coarsening step based on the triangular spectral element posteriori error estimation will be proceeded repeatedly until obtaining the optimal mesh. Some efficient precondioners will be proposed to speed up the computation. The developed method in the project will be implemented in parallel on a computer cluster with distributed memory, and will be incorporated into our existing code. Finally, the new code will be used to perform the large scale simulation of complex fluids. .. There are three essential theoretical problems to be solved in the project. One is the Inf-Sup condition of triangular spectral element when solving Navier-Stokes equations. We need choose appropriate approximate spaces for velocity and pressure to avoid suspicious pressure mode, and give a rigorous proof. Another is the posteriori error estimation of triangular spectral element for elliptic equation, which is the theoretical foundation for our adaptive method. The last one is the construction and analysis of the preconditioners, which will resort to the framework of iterative subspace decomposition. The resolution of these theoretical problems will bring the adaptive triangular spectral element method to be an important approach for the large scale simulation of complex fluids.

本项目的目标是基于三角谱元方法发展一套适用于复杂区域不可压流体模拟的自适应算法。与传统谱元方法使用四边形（六面体）单元不同，三角谱元方法可使用三角形（四面体）单元，这使得该方法更容易对网格进行细化、移动等操作，因此也更合适发展自适应算法。本项目将利用成熟的三角剖分技术处理复杂几何区域，同时借鉴自适应有限元方法中的网格细化（粗化）算法，使用最优的网格求解问题。本项目还将提出有效的预条件子和并行算法加快求解速度。本项目在理论上主要解决三角谱元的Inf-Sup 条件，后验误差估计，预条件子的构造分析等问题。这些理论问题的解决将加快三角谱元方法的发展，亦将扩大其应用范围。本项目的成果将使得三角谱元方法成为大规模复杂流体问题计算的一种重要方法。

谱方法具有高精度等优点，研究三角谱元方法并据此进一步发展自适应算法对扩大谱方法在具有复杂计算区域问题上的应用具有重要意义。本项目考察了现有的两种三角形上的谱方法，将其推广至非结构网格。特别地，我们实现了一种基于Modal有理基函数的三角谱元方法，应用到椭圆型、抛物型方程的求解，得到了最优的误差估计;基于此三角谱元方法，我们给出了二阶椭圆型偏微分方程上一种残差类型的后验误差估计，并据此发展了一种h类型的自适应网格加密算法，初步的数值实验验证了算法的有效性；在离散系统求解方面，我们提出了一种Schur Complement算法，该算法充分利用谱元离散系统的块对角结构和单元刚度质量矩阵的稀疏性，使得大型问题得以有效求解。在谱元方法的应用方面，我们开展了电磁波隐形材料的模拟。