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- G. Dziuk
- 1989

An Algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of Poisson problems on a series of surfaces. This gives a new method to solve Plateau's problem for H-surfaces.

- G. DZIUK, C. M. ELLIOTT
- 2005

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)

- Alan Demlow, Gerhard Dziuk
- SIAM J. Numerical Analysis
- 2007

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)

- Gerhard Dziuk
- Numerische Mathematik
- 2008

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)

- Gerhard Dziuk, Charles M. Elliott
- Math. Comput.
- 2013

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)

- Klaus Deckelnick, Gerhard Dziuk
- Numerische Mathematik
- 2002

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)

- Gerhard Dziuk, Ernst Kuwert, Reiner Schätzle
- SIAM J. Math. Analysis
- 2002

- Gerhard Dziuk, Charles M. Elliott
- Computat. and Visualiz. in Science
- 2010

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)

- Gerhard Dziuk, John E. Hutchinson
- Math. Comput.
- 1999

We solve the problem of finding and justifying an optimal fully discrete finite element procedure for approximating minimal, including unstable, surfaces. In a previous paper we introduced the general framework and some preliminary estimates, developed the algorithm and give the numerical results. In this paper we prove the convergence estimate.

In this article we define a surface finite element method (SFEM) for the numerical solution of parabolic partial differential equations on hypersurfaces Γ in R. The key idea is based on the approximation of Γ by a polyhedral surface Γh consisting of a union of simplices (triangles for n = 2, intervals for n = 1) with vertices on Γ. A finite element space of… (More)