流形上之区域分解法

Numerically solving PDEs on manifolds is an advanced research area which is having been intensively studied. This subject has applications in many areas such as computer graphics and astrophysics. However, the main methods in this subject have many shortcomings. In this project, we try to propose a method which overcomes some of those shortcomings. The basic idea of our approach is as follows: using domain decomposition methods (DDMs), a problem on a manifold is converted to sub-problems in subdomains, and the sub-problems are in turn converted to problems in Euclidean spaces. Many topics related to our approach have not been studied because this approach is brand-new. We will try to study these topics by working on three aspects of this project, which are: (1) applying optimized Schwarz DDM to problems on manifolds, (2) applying additive Schwarz DDM to problems on manifolds, (3) problems in high dimensional Euclidean spaces.

数值求解流形上的偏微分方程是前沿研究领域，研究成果深刻。该学科有广泛应用范围，如计算机图形学、天体物理等。但此学科中的通行计算方法有诸多弊端。本项目试图提出一个方法，来克服彼等某些弊端。我们的基本思想如下：用区域分解法，将流形上的问题化为子区域中的子问题，再将子问题化为欧氏空间中的问题。由于我们的方法是全新的，它涉及的很多课题有待研究。本项目试图从三方面入手，来研究这些课题。这三方面是：(1) 用优化Schwarz区域分解法解流形上的问题，(2) 用加性Schwarz区域分解法解流形上的问题，(3) 高维欧氏空间中的问题。

数值求解流形上的偏微分方程是前沿研究领域，研究成果深刻。该学科有广泛应用范围，如计算机图形学、天体物理等。但此学科中的通行计算方法有诸多弊端。本项目的初衷是提出一个方法，来克服彼等某些弊端。而今我们提出了一个行之有效的流形上求解二阶椭圆方程的数值区域分解法。该法是全新的数值方法，完全避开了流形上的整体网格剖分这一难题。我们在高维复杂流形上做了数值实验，效果优异。因此我们在根本上实现了本项目的初衷。另外，我们研究了相关的流形理论，为该方法的今后发展奠定基础。

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