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.
