Convergence of a fully discrete variational scheme for a thin-film equation
Provides rigorous convergence guarantees for a numerical scheme for a fourth-order PDE, addressing a known challenge in numerical analysis for gradient flows.
This paper proves convergence of a fully discrete Lagrangian scheme for the thin-film equation (Hele-Shaw flow) in 1D, leveraging gradient flow structure and dissipation properties to establish convergence to weak solutions without CFL conditions.
This paper 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 $L^2$-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-to-continuous limit. Thanks to the time-implicit character of the scheme, no CFL-type condition is needed. Numerical experiments illustrate the practicability of the scheme.