An Eulerian finite element method for partial differential equations posed on surfaces

本研究项目的目标是开发和分析一种新的欧拉有限元方法，用于求解超曲面上定义的椭圆型和抛物型偏微分方程。该方法利用体积有限元空间函数在某一曲面上的轨迹，对该曲面上的方程进行离散化。本项目旨在将该方法及其分析和应用扩展到以下几个方面：(i)将该方法扩展和分析到一个不断变化的表面；这是在时空有限元方法的框架下完成的；（二）发展高阶曲面有限元法；这涉及到超曲面上高阶有限元空间轨迹的性质分析；（iii）用体积和表面有限元方法离散的一类体域-表面耦合问题的误差分析；这包括数值分析和实验问题的平衡两相不可压缩粘流与表面活性剂（表面活性剂）。表面上的偏微分方程出现在许多自然现象的数学模型中：沿晶界扩散，生物膜中的脂质相互作用，模式形成，多相流界面上表面活性剂的运输等等。数值模拟在更好地理解和预测涉及这些或其他表面现象的过程中起着重要作用。虽然曲面方程的数值方法研究是一个快速发展的研究领域，但演化曲面和隐式曲面的计算技术在很大程度上还处于起步阶段。该项目开发的数值方法是基于连续介质运动的欧拉描述。选择欧拉描述而不是拉格朗日描述是基本的。它带来了严重的算法和分析挑战，但它与计算力学中的大多数方法一致，因此可以将该方法集成到许多现有的科学计算软件包中。该项目具体应用的一个例子是两相不可压缩流动问题中表面活性剂在界面上的传输。在这种应用中，表面（两种不同流体之间的界面，如水和油）在大量流体流动的驱动下演变。为了解释变化的表面张力现象，如马兰戈尼力，人们必须求解表面活性剂浓度在演化表面上的输运-扩散方程。可靠的流体界面模拟过程的计算工具对于严格理解这种非常复杂的两相流问题的行为至关重要。

The goal of this research project is the development and analysis of a new Eulerian finite element method for solving elliptic and parabolic partial differential equations defined on hypersurfaces. The method uses traces of volume finite element space functions on a surface to discretize equations posed on that surface. This project aims at extending the method, its analysis and applications in several directions: (i) The extension and the analysis of the method for the case of an evolving surface; This is done in the framework of space-time finite element methods; (ii) The development of a higher order surface finite element method; This involves the analysis of the properties of traces of higher order finite element spaces on hypersurfaces; (iii) An error analysis for a class of coupled bulk domain - surface problems, discretized with volume and surface finite element methods; This includes numerical analysis and experiments for the problem of equilibrium two-phase incompressible viscous flow with surface active agents (surfactants).Partial differential equations posed on surfaces arise in mathematical models for many natural phenomena: diffusion along grain boundaries, lipid interactions in biomembranes, pattern formation, and transport of surfactants on multiphase flow interfaces to mention a few. Numerical simulations play an important role in a better understanding and prediction of processes involving these or other surface phenomena. Although, the study of numerical methods for equations on surfaces is a rapidly growing research area, computational technique for evolving and implicitly defined surfaces is largely in its infant stage. Numerical methods developed in the project are based on the Eulerian description of the motion of continuous medium. This choice of the Eulerian instead of the Lagrangian description is fundamental. It leads to serious algorithmic and analysis challenges, but it is consistent with most of approaches in computational mechanics and so enables an integration of the method in many existing software packages for scientific computing. One example of a specific application the project aims is the transport of surface active agents on the interface in two-phase incompressible flow problems. In this application, the surface (interface between two different fluids, such as water and oil) evolves driven by a bulk fluid flow. To account for variable surface tension phenomena, such as Marangoni forces, one has to solve transport-diffusion equations for surfactant concentration on the evolving surface. Reliable computational tools for simulation processes on fluidic interfaces are crucial for a rigorous understanding of the behaviour of such very complex two-phase flow problems.