The original content of this thesis consists of three chapters. The first extends the surface finite element method to problems where a diffusion on a surface is coupled to diffusion in a bulk domain. The second studies an evolving surface finite element method applied to a Cahn-Hilliard equation. Finally, the third looks at new unfitted finite element methods for surface partial differential equations.
The first problem we tackle, shown inChapter 2, is a coupled bulk-surface equation. Often, applications of the surface finite element method consider problems where the evolution of surface concentrations depend on ‘bulk effects’. These effects fall into two broad categories. In each, we assume a surface is embedded in a volumetric domain. In the first case, the substance which lives on the surface may also live in parts of the volumetric region. The second case considers an evolution of the surface forced by some underlying equations for motion in the surrounding volume. Of course, both effects can occur in the same model.
To develop a method for these applications, we consider the following model problem. Given a domain Ω with closed boundary Γ, we seek a solution pairu: Ω→
Rand v: Γ→Rsatisfying −∆u+u=f in Ω (1.4.1a) (αu−βv) +∂u ∂ν = 0 on Γ (1.4.1b) −∆Γv+v+ ∂u ∂ν =g on Γ. (1.4.1c)
We assume f: Ω → R and g: Γ → R are given functions and α, β are positive constants. Equations (1.4.1a) and (1.4.1c) represent diffusion equation in the bulk and on the surface, and (1.4.1b) represents the exchange of concentration between the bulk and surface phases. This particular choice of coupling on the surface has been used by Novak et al. (2007). It can be viewed as a linearisation of the more general equation
L(u, v) +∂u
∂ν = 0,
where∂uL(u, v)>0 and ∂vL(u, v)<0, which has been used by: Kwon and Derby (2001); Booty and Siegel (2010); Medvedev and Stuchebrukhov (2011); and R¨atz and R¨oger (2012), for example. We leave the numerical analysis of more general couplings, the parabolic case and evolving domains to future work.
Our method works by taking a polyhedral approximation Ωh of Ω and using the boundary faces of Ωh, which we will call Γh, as an approximation of Γ. We then use a finite element method to solve a variational form of the above equations. This work also includes the use of higher order isoparametric finite elements. We show well-posedness for these equations and derive optimal order error estimates for the finite element method. This chapter also includes details of a numerical implementation and examples to demonstrate the rates of convergence.
In Chapter 3, we consider our second problem looking at a Cahn-Hilliard equation on an evolving surface. The Cahn-Hilliard equation (Cahn and Hilliard
1958) can be used to model several natural phenomena; applications using a Cahn- Hilliard equation can be found in Section 1.3. Analysis of the Cahn-Hilliard equa- tion, in planar domains, started in the 1980’s with the work ofElliott and Songmu
(1986) and numerical work of Elliott and French (1987, 1989) and Elliott, French and Milner (1989), which was extended to stationary surfaces byDu et al. (2011). A review of the behaviour of the Cahn-Hilliard equation in the planar case is given byElliott(1989).
We will study the following problem mathematically. We assume we are given an evolving surface{Γ(t)}, fort∈[0, T], with prescribed velocity v. We seek a solutionu of ∂•u+u∇Γ·v= ∆Γ −ε∆Γu+ 1 εψ 0(u) on Γ(t), (1.4.2)
subject to the initial condition u(·,0) = u0 on Γ(0) = Γ0. This is a fourth-order
non-linear equation posed on an evolving domain. We will look for solutions via a second-order splitting method. We assume that ε is a small, but fixed, positive parameter andψis the quartic double-well potential given by
ψ(z) = 1 4(z
2−1)2.
This is taken as a simplification of the logarithmic potential (1.3.1). We note that for general surface evolutions the Ginzburg-Landau functional will not decrease along solutions of this equation and (1.4.2) is not a gradient flow. One can enforce energy decrease by imposing extra assumptions on v. Alternatively, one can calculate a coupled gradient flow equation foruandvusing techniques fromElliott and Stinner
(2010a).
This chapter is broken into four sections. In the first we derive the continuous equation above (1.4.2). This comes from a simple conservation law on a surface and applying a generalisation of the Reynolds transport theorem to curved surfaces. Next, we derive our evolving surface finite element method. This is based on the original method ofDziuk and Elliott(2007a). In section four, the discrete solution is shown to satisfy an energy bound, hence we can use weak convergence results, along with domain perturbation arguments, to show that the continuous equations have a solution. The fifth section then shows that the finite element method converges with optimal order errors in appropriate surface norms. The chapter finishes with various numerical examples confirming the analytical results.
The final problem we consider, shown in Chapter 4, studies unfitted finite element methods for surface partial differential equations. We suppose we are given
a level set function describing the surface which may have been obtained using a level set or phase field method for a geometric partial differential equation. This method has the possibility of use in a large variety of applications where volumetric forces determine the position and geometry of an evolving interface. We would like a method with the efficiency of the parametric approach of the surface finite element method, but without worrying about constructing a good triangulation from an implicit representation of the surface. Our starting points are the sharp interface method ofOlshanskii et al.(2009) and the narrow band method ofDeckelnick et al.
(2010). We extend these methods by using the full Cartesian gradient of basis functions instead of projecting onto the tangential directions to the surface.
Given a smooth level set function Φ with Γ = {x ∈ Rn+1 : Φ(x) = 0}, we
wish to solve the surface elliptic problem:
−∆Γu+u=f on Γ. (1.4.3)
We assume we have a fixed bulk triangulation Th of a neighbourhood of Γ and Φh is some approximation of Φ (the nodal interpolant, for example). We define Γh := {x ∈ Rn+1 : Φh(x) = 0} and Dh := {x ∈ Rn+1 : |Φh(x)| < h}, which both consist of partial elements. For the sharp interface method, we set Vh to be the space of piecewise linear finite element functions over the set of elements inTh which intersect Γh (plus some technical assumptions) and solve
Z Γh ∇uh· ∇φh+uhφhdσh= Z Γh feφhdσh for allφh ∈Vh. (1.4.4)
Alternatively, for the narrow band method, we setVh to be the space of piecewise linear finite element functions over the set of elements in Th which intersects Dh and solve 1 2h Z Dh ∇uh· ∇φh+uhφhdx= 1 2h Z Dh feφhdx for allφh ∈Vh. (1.4.5)
The use of full gradients means we no longer have degenerate equations to solve and gives us control over the error of our finite element method away from the surface since we can bound the gradient of the error in the normal direction to the surface. The properties of these new methods are explored both analytically and numerically for a surface Poisson equation (1.4.3) and are shown to give comparable results to the surface finite element method.
The thesis is completed by Appendix A which sets up our notation and assumptions. This preliminary material describes the surface finite element method
as proposed byDziuk (1988). Many of the proofs from main chapters are given in full detail here taken from Dziuk and Elliott (2013b). This section also includes details of numerical experiments which will be used as a basis for comparison for the other chapters.