Optimal error estimates of the local discontinuous Galerkin method for surface diffusion of graphs on Cartesian meshes (Q427218)

The authors consider the local discontinuous Galerkin (LDG) method for surface diffusion of graphs \[ \begin{cases} u_t+\nabla \left( Q \left( I-\frac{\nabla u \otimes \nabla u }{Q^2}\right) \nabla H \right)=0 &\text{for }(x.t) \in \Omega \times (0,T] ,\\ u(x,0)=u_0(x) &\text{in } \Omega,\end{cases} \] subject to periodic boundary conditions, where \(\Omega\) is a bounded rectangular domain in \(\mathcal{R}^d, d \leq 3\), \(Q:=\sqrt{1+|\nabla u|^2}\) is the area element and \(H:= \nabla \cdot (\nabla u/Q)\) is the mean curvature of the domain boundary. The LDG method for surface diffusion of graphs was developed in an earlier paper by \textit{Y. Xu} and \textit{C.-W. Shu} [J.\ Sci.\ Comput.\ 40, No. 1--3, 375--390 (2009; Zbl 1203.65190)], where also its energy stability was proved. In the present paper, a priori error estimates are analyzed and the optimal convergence order \(k+1\) in the \(L^2\) norm is obtained when using completely discontinuous polynomials of degree \(k\) as trial functions.