This review concerns the computation of curvature-dependent interface motion governed by geometric partial differential equations. The canonical problem of mean curvature flow is that of finding a surface which evolves so that, at every point on the surface, the normal velocity is given by the mean curvature. In recent years the interest in geometric PDEs involving curvature has burgeoned. Examples of applications are, amongst others, the motion of grain boundaries in alloys, phase transitions and image processing. The methods of analysis, discretization and numerical analysis depend on how the surface is represented. The simplest approach is when the surface is a graph over a base domain. This is an example of a sharp interface approach which, in the general parametric approach, involves seeking a parametrization of the surface over a base surface, such as a sphere. On the other hand an interface can be represented implicitly as a level surface of a function, and this idea gives rise to the so-called level set method. Another implicit approach is the phase field method, which approximates the interface by a zero level set of a phase field satisfying a PDE depending on a new parameter. Each approach has its own advantages and disadvantages. In the article we describe the mathematical formulations of these approaches and their discretizations. Algorithms are set out for each approach, convergence results are given and are supported by computational results and numerous graphical figures. Besides mean curvature flow, the topics of anisotropy and the higher order geometric PDEs for Willmore flow and surface diffusion are covered.

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Abstract. We analyze a semidiscrete numerical scheme for approximating the evolution of parametric curves by elastic flow in R n . The fourth order equation is split into two coupled second order problems, which are approximated by linear finite elements. We prove error bounds for the resulting scheme and present numerical test calculations that confirm our analysis.

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In this paper, we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements. The key idea is that the n-dimensional hypersurface, Γ ⊂ R n+1 , is embedded in a polyhedral domain in R n+1 consisting of a union, T h , of (n + 1)-simplices. The finite element approximating space is based on continuous piece-wise linear finite element functions on T h . Our first method is a sharp interface method, SIF, which uses the bulk finite element space in an approximating weak formulation obtained from integration on a polygonal approximation, Γ h , of Γ. The full gradient is used rather than the projected tangential gradient and it is this which distinguishes SIF from the method of [42]. The second method, NBM, is a narrow band method in which the region of integration is a narrow band of width O(h). NBM is similar to the method of [13] but again the full gradient is used in the discrete weak formulation. The a priori error analysis in this paper shows that the methods are of optimal order in the surface L 2 and H 1 norms and have the advantage that the normal derivative of the discrete solution is small and converges to zero. Our third method combines bulk finite elements, discrete sharp interfaces and narrow bands in order to give an unfitted finite element method for parabolic equations on evolving surfaces. We show that our method is conservative so that it preserves mass in the case of an advection diffusion conservation law. Numerical results are given which illustrate the rates of convergence.

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We consider the variational discretization of elliptic Dirichlet optimal control problems with constraints on the control. The underlying state equation, which is considered on smooth twoand three-dimensional domains, is discretized by linear finite elements taking into account domain approximation. The control variable is not discretized. We obtain optimal error bounds for the optimal control in two and three space dimensions and prove a superconvergence result in two dimensions provided that the underlying mesh satisfies some additional condition. We confirm our analytical findings by numerical experiments.

We consider an elliptic optimal control problem with control constraints and pointwise bounds on the gradient of the state. We present a tailored finite element approximation to this optimal control problem, where the cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of the lowest order Raviart-Thomas mixed finite element. Pointwise bounds on the gradient variable are enforced in the elements of the triangulation. Controls are not discretized. Error bounds for control and state are obtained in two and three space dimensions. A numerical example confirms our analytical findings.