In (Xu and Shu in J. Sci. Comput. 40:375-390, 2009), a local discontinuous Galerkin (LDG) method for the surface diffusion of graphs was developed and a rigorous proof for its energy stability was given. Numerical simulation results showed the optimal order of accuracy. In this subsequent paper, we concentrate on analyzing a priori error estimates of the LDG method for the surface diffusion of graphs. The main achievement is the derivation of the optimal convergence rate k+1 in the L 2 norm in one-dimension as well as in multi-dimensions for Cartesian meshes using a completely discontinuous piecewise polynomial space with degree k≥1.
Bibliographical noteFunding Information:
Research of Y. Xu was supported by NSFC grant No. 10971211, No. 11031007, FANEDD No. 200916, FANEDD of CAS, NCET No. 09-0922 and the Fundamental Research Funds for the Central Universities.
- Error estimates
- Local discontinuous Galerkin method
- Surface diffusion of graphs