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An unconditionally stable finite element scheme for anisotropic curve shortening flow

Klaus DeckelnickRobert Nürnberg — 2023

Archivum Mathematicum

Based on a recent novel formulation of parametric anisotropic curve shortening flow, we analyse a fully discrete numerical method of this geometric evolution equation. The method uses piecewise linear finite elements in space and a backward Euler approximation in time. We establish existence and uniqueness of a discrete solution, as well as an unconditional stability property. Some numerical computations confirm the theoretical results and demonstrate the practicality of our method.

Discrete anisotropic curvature flow of graphs

Klaus DeckelnickGerhard Dziuk — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

The evolution of –dimensional graphs under a weighted curvature flow is approximated by linear finite elements. We obtain optimal error bounds for the normals and the normal velocities of the surfaces in natural norms. Furthermore we prove a global existence result for the continuous problem and present some examples of computed surfaces.

Optimal error Estimates for the Stokes and Navier–Stokes equations with slip–boundary condition

Eberhard BänschKlaus Deckelnick — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a finite element discretization by the Taylor–Hood element for the stationary Stokes and Navier–Stokes equations with slip boundary condition. The slip boundary condition is enforced pointwise for nodal values of the velocity in boundary nodes. We prove optimal error estimates in the and norms for the velocity and pressure respectively.

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