Algorithms for approximating continuum processes on surfaces

连续过程在应用科学和自然科学中无处不在。由于解析解很少是可能的，相应的偏微分方程（PDE）模型的准确和有效的算法的实际重要性，不能过分强调。已经有了很大的努力，发展的许多重要类的偏微分方程的数值方法。这些努力侧重于在一个、两个或三个空间维度上找到解决办法。但涉及一般微分方程的问题也出现在二维曲面或一维弯曲的细丝上。例如，表面上的PDE可用于在计算机生成的表面上放置纹理，或用于增强或恢复扫描表面上的损坏图案。 在物体的机器识别中，表面偏微分方程的解可以用来表征物体的形状。 在材料科学中，这样的方程已被用于检查曲面上材料的相变，而在理论生物学中，表面上的PDE作为建模骨病理学（如骨质疏松症）的一部分而出现。 尽管在曲面上的偏微分方程的广泛发生，仍然需要一个系统的方法来有效地计算一般曲面上的一般偏微分方程的解决方案。 我们的研究将继续发展和分析的算法求解偏微分方程的表面。重点是方法，（1）适用于一般的表面，（2）适用于一般的偏微分方程，（3）是准确和有效的，（4）是简单的，在这个意义上，他们对待不同的偏微分方程在统一的方式，利用标准的，现有的技术和软件。预计在这项研究资助下获得的方法将使研究人员和最终用户能够以极高的精度对一般表面过程的现实模型进行数值研究。

Continuous processes are ubiquitous throughout the applied and natural sciences. Because analytical solutions are rarely possible, the practical importance of accurate and efficient algorithms for the corresponding partial differential equation (PDE) models 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 two-dimensional curved surfaces or one-dimensional curved filaments. For example, PDEs on surfaces can be used to place a texture on a computer generated surface, or to enhance or restore a damaged pattern on a scanned surface. In machine recognition of objects, the solution of surface PDEs can be used to characterize the shapes of objects. In material science such equations have been used to examine phase change of a material on a curved surface, while in theoretical biology, PDEs on surfaces arise as part of the modeling bone pathologies such as osteoporosis. Despite the widespread occurrence of PDEs on curved surfaces there is still a need for a systematic approach for effectively computing the solution of general PDEs on general surfaces. Our research will continue the development and analysis of algorithms for solving PDEs on surfaces. The focus is on methods which (1) apply to general surfaces, (2) apply to general PDEs, (3) are accurate and efficient, and (4) are simple in the sense that they treat different PDEs in uniform fashion by making use of standard, existing techniques and software. It is anticipated that the methods obtained under this research grant will enable researchers and end-users to numerically investigate realistic models of general surface processes with great accuracy.