曲面上偏微分方程可用来描述物理和生物科学中很多有趣现象，其在数学性质和计算方法上与普通偏微分方程非常不同，最近在数值分析领域引起很大兴趣。虽然针对某些具体问题，人们已经发展出多种方法，但是曲面上偏微分方程的数值方法还很不成熟。基于已有研究，我们拟发展复杂曲面上非线性偏微分方程的高效数值方法、曲面上的高阶Trace有限元方法、以及曲面偏微分方程与曲面运动几何偏微分方程耦合系统的数值方法。我们还将结合数值分析方法与曲面的几何理论，构造新的计算和分析工具。通过这项研究，不仅可满足物理和生物相关现象模拟的实际需要，还可为科学计算和数值分析领域增添新的研究内容。

Many phenomena in physics and biology can be modelled by partial differential equations(PDEs) on surfaces. Surface PDEs have different properties from the standard ones both in mathematics and numerics. Recently, they have arisen much interest in numerical analysis. Based on our previous work, we will develop some efficient numerical methods for nonlinear partial differential equations on complicated surfaces. We will develop high order Trace finite element methods. We will also compute the coupling system of surface PDEs and geometric PDEs for motion of the surface. We would like to develop new analytic tools by combing traditional numerical analysis methods with the geometric theory of surface and manifold. Our results will be used to simulate some interesting problems in physics and biology. This will also provide some new contents to scientific computation and numerical analysis.