Numerical methods for solving partial differential equations on surfaces

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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.