Int. J. Numer. Anal. Mod., 8 (2011), pp. 252-283. Optimal error estimates of the local discontinuous Galerkin method for Willmore flow of graphs on Cartesian meshes L. Ji 1, Y. Xu 11 Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China. Received by the editors September 25, 2009 and, in revised form, August 8, 2010. Abstract In this paper, we analyze a local discontinuous Galerkin method for the willmore flow of graphs. We derive the optimal error estimates for this nonlinear equation in one-dimension and in multi-dimensions for Cartesian meshes using completely discontinuous piecewise polynomial space with degree $k \leq 1$. AMS subject classifications: 65M60, 35K55 Key words: local discontinuous Galerkin method, Willmore flow of graphs, stability, error estimates. Email: jlyue@mail.ustc.edu.cn, yxu@ustc.edu.cn