A simple embedding method for solving partial differential equations on surfaces

Abstract

It is increasingly common to encounter partial differential equations (PDEs) posed on surfaces, and standard numerical methods are not available for such novel situations. Herein, we develop a simple method for the numerical solution of such equations which embeds the problem within a Cartesian analog of the original equation, posed on the entire space containing the surface. This allows the immediate use of familiar finite difference methods for the discretization and numerical solution. The particular simplicity of our approach results from using the closest point operator to extend the problem from the surface to the surrounding space. The resulting method is quite general in scope, and in particular allows for boundary conditions at surface boundaries, and immediately generalizes beyond surfaces embedded in ${\mathbb{R}}^3$, to objects of any dimension embedded in any ${\mathbb{R}}^n$. The procedure is also computationally efficient, since the computation is naturally only carried out on a grid near the surface of interest. We present the motivation and the details of the method, illustrate its numerical convergence properties for model problems and also illustrate its application to several complex model equations.

Introduction

Many applications in the natural and applied sciences require the solutions of partial differential equations (PDEs) on surfaces or more general manifolds. Examples of such application areas arise in biological systems, image processing, medical imaging, mathematical physics, fluid dynamics and computer graphics. For example, applications in image processing include the generation of textures [27], [28], the visualization of vector fields [7] and weathering [8], while applications in fluid dynamics include flows and solidification on surfaces [17], [18], and the problem of evolving surfactants on interfaces [29].

A popular approach to solving PDEs on surfaces is to impose a smooth coordinate system or parameterization on the surface, express differential operators within these coordinates, and then discretize the resulting equations. See, for example [10] for a tutorial and survey of methods for parameterizing surfaces. However, the required coordinates can be complicated or impractical to construct, and the coordinates may introduce artificial singularities, such as at the poles in spherical coordinates. Indeed, as pointed out in [10], “parameterizations almost always introduce distortion in either angles or some region”. Also, equations that are simple when written using intrinsic derivatives, such as surface diffusion, become substantially more complex when written in a coordinate system, involving nonconstant coefficients and more derivative terms.

Another common approach to solving partial differential equations on surfaces is to solve the PDE directly on a triangulation of the surface. This approach can be effective for certain classes of equations, however, as a general technique it leads to a number of difficulties. These are discussed in [3], [4]. In particular, this approach leads to nontrivial discretization procedures for the differential operators, as well as difficulties in accurately computing geometric primitives, such as surface normals and curvature [3]. In addition, convergence of numerical schemes on triangulated surfaces remains less well understood than methods on Cartesian grids [11].

An alternative approach to treating PDEs on surfaces is to embed the surface differential equations of interest within differential equations posed on all of ${\mathbb{R}}^3$, so that the solution of the embedding equations, when restricted to the surface, provides the solution to the original problem. With such an approach, the ultimate goal is to develop a method that allows the treatment of general, complex surface geometry, while still retaining the simplicity that comes from working in standard Cartesian coordinates. With this in mind, Bertalmío et al. [3] introduced an embedding method for solving variational problems and the resulting Euler–Lagrange evolution PDEs on surfaces. In their approach, the underlying surface is represented as the level-set of a higher dimensional function and the evolution corresponding to the surface PDEs is carried out via PDEs that are posed on all of ${\mathbb{R}}^3$. This leads to equations that can be discretized and solved using Cartesian grid methods. A further improvement to this method was proposed by Greer [11]. In his approach, the evolution equation is modified away from the surface to maintain greater regularity of the solution near the surface of interest. See also [1] for related work on the finite element approximation of elliptic partial differential equations on implicit surfaces via level-set methods.

Embedding methods based on level-set methods have a number of limitations, however. Most obviously, these methods do not immediately allow for open surfaces with boundaries, or filamentary objects of codimension-two or higher, although it is in principle possible to represent such objects by introducing additional level-set functions. Another limitation is that these methods result in an embedding PDE posed on all of space, and complications arise when they are solved in a restricted computational band around the surface. Such “narrow banding” requires the imposition of appropriate boundary conditions, and how to best impose these conditions is not generally understood. For example, even using the regularity improvements proposed by Greer [11], a degradation of the order of convergence is observed when banding is used in diffusive problems. At a more technical level, level-set based methods also either lead to degenerate diffusion equations or require the use of an additional diffusion step when applied to parabolic equations [11].

Similar to other embedding methods, the approach we present here discretizes the partial differential equations using a fixed Cartesian grid in the embedding space. However, our method is based on the use of a closest point representation of the surface (cf. [16]) rather than a level-set representation. In conjunction with this change in representation, we abandon the concept of solving an embedding PDE for all time, and instead use the embedding PDE to advance the solution near the surface for one time step (or one stage of a Runge–Kutta method). This leads to a new method for solving PDEs on surfaces which has great simplicity, as well as additional desirable features. Most importantly, the embedding PDE is the obvious analog of the surface PDE, and involves only the standard Cartesian differential operators. In addition, the method can treat open surfaces and is not limited to objects of codimension-one. It also naturally allows the computation to be done on a grid defined in a narrow band near the surface without any degradation in the order of accuracy and without imposing artificial boundary conditions. To distinguish our approach from other embedding procedures, and to emphasize its essential reliance on the closest point representation, we refer to the method as the closest point method. As an aside, note that a special case of this procedure was introduced in our recent paper on the diffusion-generated motion of curves on surfaces [16]. It was in this context that the general potential of the approach was first realized, although it occurs only as a special, simple procedure for approximating in-surface curvature motion according to the diffusion-generated motion algorithm.

The paper unfolds as follows. Section 2 gives the method, describes its implementation and provides an analysis. In Section 3, we give a number of two-dimensional convergence studies to validate the method. Section 4 considers a variety of three-dimensional convergence tests and provides some examples that are relevant to biology: a Fitzhugh–Nagumo and a morphogenesis simulation. In Section 5, we give a summary and discuss some of our ongoing projects in the subject. Finally, Appendix A concludes with a discussion on the computation of geometric quantities defined on the surface.