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。由于解析解是不可能的，精确和有效的算法对相应的偏微分方程（PDE）模型的实际重要性怎么强调也不为过。对于许多重要的偏微分方程的数值方法，人们已经付出了很大的努力。这些努力的重点是在一个、两个或三个空间维度上寻找解决方案。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，或在扫描表面上增强或恢复损坏的图案。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，这样的方程被用来检查材料在曲面上的相变，而在理论生物学中，表面上的偏微分方程作为骨质疏松症等骨病理模型的一部分出现。尽管曲面上普遍存在偏微分方程，但仍然需要一种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。

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.