Gerhard Dziuk

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In this article, we define a new evolving surface finite-element method for numerically approximating partial differential equations on hypersurfaces Γ (t) in Rn+1 which evolve with time. The key idea is based on approximating Γ (t) by an evolving interpolated polyhedral (polygonal if n = 1) surface Γh(t) consisting of a union of simplices (triangles for n(More)
We present an adaptive finite element method for approximating solutions to the Laplace-Beltrami equation on surfaces in R3 which may be implicitly represented as level sets of smooth functions. Residual-type a posteriori error bounds which show that the error may be split into a “residual part” and a “geometric part” are established. In addition,(More)
We propose a new algorithm for the computation of Willmore flow. This is the L2-gradient flow for the Willmore functional, which is the classical bending energy of a surface. Willmore flow is described by a highly nonlinear system of PDEs of fourth order for the parametrization of the surface. The spatially discrete numerical scheme is stable and(More)
In this paper we consider the evolving surface finite element method for the advection and diffusion of a conserved scalar quantity on a moving surface. In an earlier paper using a suitable variational formulation in time dependent Sobolev space we proposed and analysed a finite element method using surface finite elements on evolving triangulated surfaces.(More)
We analyze a fully discrete numerical scheme approximating the evolution of n–dimensional graphs under anisotropic mean curvature. The highly nonlinear problem is discretized by piecewise linear finite elements in space and semi–implicitly in time. The scheme is unconditionally stable und we obtain optimal error estimates in natural norms. We also present(More)
In this article we define a level set method for a scalar conservation law with a diffusive flux on an evolving hypersurface Γ (t) contained in a domain Ω ⊂ Rn+1. The partial differential equation is solved on all level set surfaces of a prescribed time dependent function Φ whose zero level set is Γ (t). The key idea lies in formulating an appropriate weak(More)