Convergence of a finite element method for non-parametric mean curvature flow

Summary. Convergence for the spatial discretization by linear finite elements of the non-parametric mean curvature flow is proved under natural regularity assumptions on the continuous solution. Asymptotic convergence is also obtained for the time derivative which is proportional to mean curvature. An existence result for the continuous problem in adequate spaces is included. Received September 30, 1993     

Numerical analysis of an inverse problem for the eikonal equation

We are concerned with the inverse problem for an eikonal equation of determining the speed function using observations of the arrival time on a fixed surface. This is formulated as an optimisation problem for a quadratic functional with the state equation being the eikonal equation coupled to the so-called Soner boundary condition. The state equation is discretised by a suitable finite difference scheme for which we obtain existence, uniqueness and an error bound. We set up an approximate optimisation problem and show that a subsequence of the discrete mimina converges to a solution of the continuous optimisation problem as the mesh size goes to zero. The derivative of the discrete functional is calculated with the help of an adjoint equation which can be solved efficiently by using fast marching techniques. Finally we describe some numerical results.     

Unstable Willmore surfaces of revolution subject to natural boundary conditions

In the class of surfaces with fixed boundary, critical points of the Willmore functional are naturally found to be those solutions of the Euler-Lagrange equation where the mean curvature on the boundary vanishes. We consider the case of symmetric surfaces of revolution in the setting where there are two families of stable solutions given by the catenoids. In this paper we demonstrate the existence of a third family of solutions which are unstable critical points of the Willmore functional, and which spatially lie between the upper and lower families of catenoids. Our method does not require any kind of smallness assumption, and allows us to derive some additional interesting qualitative properties of the solutions.     

Finite element approximation of elliptic control problems with constraints on the gradient

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.     

Discrete concepts for elliptic optimal control problems with constraints on the gradient of the state

We consider elliptic optimal control problems with constraints on the gradient of the state and propose two distinguish concepts for their discretization. The first concept uses piecewise linear, continuous finite element Ansatz functions for the state, while the second concept uses the lowest order Raviart–Thomas mixed finite element. In both cases variational discretization from [5] is used for the controls. We present optimal finite element error estimates for the numerical solutions and confirm our theoretical findings by a numerical experiment. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical analysis and algorithms in control and state constrained optimization with PDEs

We consider an elliptic optimal control problem with control and pointwise state constraints. The cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of linear finite elements and enforcing the state constraints in the nodes of the triangulation. The control variable is not discretized. A general error bound for control and state is obtained which forms the starting point for optimal error estimates in both in two and three space dimensions. For the numerical implementation of the discrete concept fix-point iterations or generalized Newton methods are proposed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on the approximation of elliptic control problems with bang-bang controls

In the present work we use the variational approach in order to discretize elliptic optimal control problems with bang-bang controls. We prove error estimates for the resulting scheme and present a numerical example which supports our analytical findings.