Development and analysis of an Eulerian finite element method for partial differential equations on implicitly defined surfaces
Final Report Abstract
In the past decade there has been a strong growth in the scientific research on the numerical analysis of methods for solving partial differential equations on surfaces. In previous joint work of the applicants, a new Eulerian finite element method for the discretization of elliptic partial differential equations on stationary surfaces is introduced. In this project we extended the analysis and application of this method in several directions. One topic that we addressed is the development and analysis of a residual based local error estimator, which can be used as basis for an adaptive variant of this finite element method. A further topic is the development of a stabilization technique such that the method is applicable to advection-diffusion problems in which the advection strongly dominates the diffusion. For this we introduced and analyzed a surface variant of the streamline-diffusion stabilization method, known from finite element discretizations in Euclidean space. As a byproduct of these investigations we derived a new geometric property of the zero-level of a function that is continuous and piecewise linear with respect to a shape regular tetrahedral triangulation. Such a zero level is a triangulation which in general contains triangles with arbitrary small interior angles. We showed that a maximal angle property holds: the maximum of the interior angles of the triangles is bounded by a constant smaller than π, which depends only on the shape regularity of the outer tetrahedral triangulation. A main result in the project is an extension of the Eulerian surface finite element method to a method for the discretization of partial differential equations on evolving surfaces. The key idea is to consider the evolving surface as a stationary manifold in a space-time domain. In this spacetime domain in R4 we use a Euclidean volume mesh consisting of four-dimensional prisms (3D tetrahedron × time inverval). The traces of standard finite element functions on these prisms are used to define surface finite element spaces. This is a new approach, which has not been studied in the literature before. In the project we studied several topics related to this spacetime technique, ranging from analysis of well-posedness of the underlying weak formulation of the partial differential equation and stability of the discrete problem, to implementation aspects. The finite element techniques developed in this project have been implemented in the DROPS software package that is used for the numerical simulation of two-phase incompressible flow problems. In such applications often so-called surfactants are present, which then give rise to advection-diffusion equations on an evolving interface. The newly developed surface finite element techniques are used in the simulation of such surfactant equations.
Publications
- An Eulerian space-time finite element method for diffusion problems on evolving surfaces, IGPM report 362, RWTH Aachen (2013)
M.A. Olshanskii, A. Reusken, X. Xu
(See online at https://dx.doi.org/10.1137/130918149) - An adaptive surface finite element method based on volume meshes, SIAM J. Numer. Anal. 50 (2012), 1624–1647
A. Demlow, M. A. Olshanksii
- A stabilized finite element method for advectiondiffusion equations on surfaces, IMA J. Numer. Anal. (2013)
M.A. Olshanskii, A. Reusken, X. Xu
(See online at https://doi.org/10.1093/imanum/drt016) - Non-degenerate Eulerian finite element method for solving PDEs on surfaces, Rus.J. Num. Anal. Math. Model. 28 (2013) 101–124
A.Y. Chernyshenko, M.A. Olshanskii
