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Optimal Error Estimates of the Local Discontinuous Galerkin Method for Surface Diffusion of Graphs on Cartesian Meshes

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, P.R. China

    Liangyue Ji & Yan Xu

  2. Delft Institute of Applied Mathematics, Delft University of Technology, 2628, CD, Delft, The Netherlands

    Liangyue Ji

Authors
  1. Liangyue Ji
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  2. Yan Xu
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Correspondence to Yan Xu.

Additional 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.

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Ji, L., Xu, Y. Optimal Error Estimates of the Local Discontinuous Galerkin Method for Surface Diffusion of Graphs on Cartesian Meshes. J Sci Comput 51, 1–27 (2012). https://doi.org/10.1007/s10915-011-9492-4

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  • DOI: https://doi.org/10.1007/s10915-011-9492-4

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