There has been burgeoning interest in the computation of partial differential equations on curves and surfaces. Models involving partial differential equations on surfaces arise in many areas including material science, bio-physics, fluid mechanics and image processing. For example, we refer to [10, 27, 29] for applications of the Allen–Cahn and Cahn–Hilliard equations to phase ordering and separation on surfaces. Models for thin fluid films on surfaces have been developed in [21, 24]. For image processing and geometry applications we mention geodesic flow of curves on surfaces and active contours for segmentation on surfaces, [6, 22, 23, 28].
The work in this paper is concerned with an approach to the formulation and approximation of parabolic equations on a prescribed stationary n-dimensional surface ? in Rn+1 (n = 1, 2) using an implicit representation of the surface. The surface is just one level set of a prescribed function ? and the partial differential equation and its solution are extended to a domain ? ? R n+1 containing the surface. A general framework for formulating partial differential equations on implicit surfaces was proposed by the authors of [3]. They considered time dependent second order linear and nonlinear diffusion equations in the context of finite difference approximations on rectangular grids independent of the surfaces. In [20, 19] the authors presented finite difference methods for fourth order parabolic equations on implicit surfaces. A finite element approximation of elliptic equations on implicit surfaces is presented in [5].
Our work is concerned with the finite element discretization of second and fourth order parabolic equations on surfaces. The idea is to solve PDEs on all level surfaces of ? in ? by discretizing a suitable variational formulation by a finite element method on a mesh which is independent of the surfaces. This defines an Eulerian formulation. Stable time stepping schemes are formulated in a natural way. By using second order splitting of the fourth order operators, H1 conforming finite element schemes can be employed for fourth order problems such as the Cahn–Hilliard equation. When the boundary of ? consists of level sets of ? it is not necessary to impose artificial boundary conditions because the triangulation is fitted to the domain ?. A remarkable feature of our numerical experiments is that, on a fixed level set, finite element approximations converge at an optimal rate. Our approach can be extended to second order diffusion problems on evolving surfaces (see [14]). See also [1] and [30]. The computing times for our method are similar to computing times for cartesian PDEs.
This approach is in contrast to approximating the PDEs directly on triangulated surfaces. In [11], [12] and [13] we introduced the surface and evolving surface finite element method (respectively SFEM and ESFEM) for the numerical solution of elliptic and parabolic equations on prescribed stationary and moving hypersurfaces. The method relies on approximating the partial differential equation on a triangulated surface (n = 2) or polygonal curve (n = 1). Naturally, where applicable, this method is more efficient than solving PDEs on implicit surfaces. On the other hand, in applications a surface might arise as a level set of a function computed from solving another coupled equation in which case the method of this paper may be attractive. Also when the surface is complex and evolving with possible topology changes it may be advantageous to employ a level set description of the surface. Finally, the method is appropriate when a PDE has to be solved on all level sets of a given function.
The layout of the paper is as follows. We begin in Section 2 by defining notation and essential concepts from elementary differential geometry necessary to describe the problem and the numerical method. The equations and variational formulations are presented in Section 3. In Section 4 the finite element method is defined. The results of numerical experiments are presented in Section 5. Finally, in Section 6 we make some concluding remarks.
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