Numerical methods for solving partial differential equations on surfaces

偏微分方程(PDE)在科学和应用科学中普遍存在。由于解析解的可能性很小，因此精确有效的数值方法对于偏微分方程组的实际重要性怎么强调都不为过。对于许多重要的偏微分方程类，人们已经做出了很大的努力来发展数值方法。这些努力的重点是寻找一、二或三个空间维度的解决方案。但在一般流形上也会出现涉及一般微分方程的问题，例如二维曲面或一维弯曲细丝。例如，在材料科学中，人们可能希望研究材料在曲面上的相变。在生物模型中，人们可以研究皮肤上的伤口愈合，或者动物皮毛上图案的进化。计算机图形和图像处理也经常在表面上使用PDE；例如，它们可能会使用这种方法在表面上放置纹理或在花瓶等表面上恢复损坏的图案。尽管曲面上的偏微分方程组广泛存在，但仍然需要一种系统的方法来精确计算一般曲面上的一般偏微分方程组的解。

Partial differential equations (PDEs) are ubiquitous throughout the sciences and applied sciences. Because analytical solutions are rarely possible, the practical importance of accurate and efficient numerical methods for PDEs cannot be overemphasized. There has been a great effort made to develop numerical methods for many important classes of PDEs. These efforts focus on finding solutions in one, two or three spatial dimensions. But problems involving general differential equations also arise on general manifolds, such as two-dimensional curved surfaces or one-dimensional curved filaments. For example, in material science one might wish to examine phase change of a material on a curved surface. In biological modeling, one might study wound healing on skin, or the evolution of a pattern on an animal coat. Computer graphics and image processing also frequently use PDEs on surfaces; for example, they might use such methods to place a texture on a surface or restore a damaged pattern on a surface such as a vase. Despite the widespread occurrence of PDEs on curved surfaces there is still a need for a systematic approach for accurately computing the solution of general PDEs on general surfaces.