Algorithms for approximating continuum processes on surfaces

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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.