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14. Convergence of a fully discrete variational scheme for a thin-film equation

From the book Topological Optimization and Optimal Transport

  • Horst Osberger and Daniel Matthes

Abstract

This chapter is concerned with a rigorous convergence analysis of a fully discrete Lagrangian scheme for the Hele-Shaw flow, which is the fourth-order thin-film equation with linear mobility in one space dimension. The discretization is based on the equation’s gradient flow structure in the L2-Wasserstein metric. Apart from its Lagrangian character - which guarantees positivity and mass conservation - the main feature of our discretization is that it dissipates both the Dirichlet energy and the logarithmic entropy. The interplay between these two dissipations paves the way to proving convergence of the discrete approximations to a weak solution in the discrete-tocontinuous limit. Thanks to the time-implicit character of the scheme, no CFL-type condition is needed.Numerical experiments illustrate the practicability of the scheme.

© 2017 Walter de Gruyter GmbH, Berlin/Munich/Boston
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