Numerical analysis and computation for partial differential equations on surfaces

本文主要研究曲面上偏微分方程的数值分析与计算。关于三维空间域上的偏微分方程有大量的研究，但是对于可能是大块三维域边界的超曲面上的微分方程的研究相对较少。这种表面偏微分方程在生物学和材料的复杂表面过程建模中越来越重要。传统上，球体方程出现在气候和天气模型中。在这种情况下，数值方法可以利用球面几何。然而，在我们所考虑的应用中（如细胞生物学、合金表面溶解、流体界面上的表面活性剂），表面的形态是任意的，可能是复杂的。在这个项目中需要解决的挑战包括：-与离散化相关的退化方程的数学分析，硬数值分析，算法的开发和应用于复杂系统。由于数学问题的新颖性和技术性以及应用的规模，该项目在范围上雄心勃勃。PDRA将与PI密切合作，解决具有挑战性的复杂问题，并将在新兴和蓬勃发展的领域接受计算应用数学的高级培训。此外，还有两名访问研究人员，他们是PI的持续合作者。该项目将产生有关新方法的定理，算法的发展和复杂物理过程的模拟。传播将在高质量的研究论文和会议上发表。

This proposal is concerned with the numerical analysis and computation of partial differential equations on surfaces. There is a vast amount of research concerned with partial differential equations posed on domains in three space dimensions, say, but relatively very little on differential equations on hypersurfaces which may be the boundaries of bulk three dimensional domains. Such surface partial differential equations are of increasing importance in modelling complex surface processes in biology and materials. Traditionally equations on spheres arise in climate and weather modelling. Numerical methods in this setting can exploit the spherical geometry. However in the applications we have in mind, (eg cell biology, alloy surface dissolution, surfactants on fluid interfaces) the morpohology of the surface is arbitrary and may be complex. The challenges to be addressed in this project include:-mathematical analysis of degenerate equations related to discretization, hard numerical analysis, development of algorithms and application to complicated systems. The project is ambitious in scope because of the novelty and technicality of the mathematical problems and the scale of the applications. A PDRA will work closely with the PI on the adventurous complex problems and will receive advanced training in computational applied mathematics in an emerging and burgeoning area. Also involved will be two visiting researchers who are ongoing collaborators of the PI. The project will result in theorems concerning new methods, the development of algorithms and simulations of complex physical processes. Dissemination will be publication in high quality research articles and talks at conferences.