A 𝐶¹–finite element method for the Willmore flow of two-dimensional graphs

Deckelnick, Klaus; Katz, Jakob; Schieweck, Friedhelm

doi:10.1090/mcom/2973

A $C^1$–finite element method for the Willmore flow of two-dimensional graphs
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Math. Comp. 84 (2015), 2617-2643 Request permission

Abstract:

We consider the Willmore flow of two-dimensional graphs subject to Dirichlet boundary conditions. The corresponding evolution is described by a highly nonlinear parabolic PDE of fourth order for the height function. Based on a suitable weak form of the equation we derive a semidiscrete scheme which uses $C^1$-finite elements and interpolates the Dirichlet boundary conditions. We prove quasioptimal error bounds in Sobolev norms for the solution and its time derivative and present results of test calculations.

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