Abstract
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.
Published Version
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