A new adaptive local mesh refinement algorithm and its application on fourth order thin film flow problem

A new adaptive local mesh refinement method is presented for thin film flow problems containing moving contact lines. Based on adaptation on an optimal interpolation error estimate in the Lp norm (1 < p ⩽ ∞) [L. Chen, P. Sun, J. Xu, Multilevel homotopic adaptive finite element methods for convection dominated problems, in: Domain Decomposition Methods in Science and Engineering, Lecture Notes in Computational Science and Engineering 40 (2004) 459–468], we obtain the optimal anisotropic adaptive meshes in terms of the Hessian matrix of the numerical solution. Such an anisotropic mesh is optimal for anisotropic solutions like the solution of thin film equations on moving contact lines. Thin film flow is described by an important type of nonlinear degenerate fourth order parabolic PDE. In this paper, we address the algorithms and implementation of the new adaptive finite element method for solving such fourth order thin film equations. By means of the resulting algorithm, we are able to capture and resolve the moving contact lines very precisely and efficiently without using any regularization method, even for the extreme degenerate cases, but with fewer grid points and degrees of freedom in contrast to methods on a fixed mesh. As well, we compare the method theoretically and computationally to the positivity-preserving finite difference scheme on a fixed uniform mesh which has proven useful for solving the thin film problem.