Optimal Convergence Analysis of a Second Order Scheme for a Thin Film Model Without Slope Selection

In Li et al. (J Sci Comput 76(3):1905–1937, 2018), a temporal second-order mixed finite element scheme has been proposed for the thin film epitaxial growth model without slope selection. Using the super-convergence theory in a regular rectangular mesh, the authors of Li et al. (2018) proved an optimal \(O(h^{q+1}+\tau ^2)\) convergence. However, in a quasi-uniform triangulation mesh setting, only a sub-optimal convergence rate \(O(h^q+\tau ^2)\) is proved, while numerical results indicated an optimal \(O(h^{q+1}+\tau ^2)\) convergence when the exact solution has \(H^{q+1}\) regularity in space. Here h and \(\tau \) are the discretization sizes in space and time, respectively, and \(q\ge 1\) is the degree of the polynomial in the spatial discretization. In this paper, we provide a theoretical proof of the optimal convergence rate. The main difficulty lies in how to treat a nonlinear term \(\frac{\nabla u}{1+|\nabla u|^2}\). We solve this by using a discrete Laplacian operator \(-\varDelta _h\) and some uncommon techniques in the analysis. Numerical results are also presented to demonstrate the \((q+1)\)-order convergence of the spatial approximation.