A Mixed Discontinuous Galerkin Method Without Interior Penalty for Time-Dependent Fourth Order Problems

Hailiang Liu Mathematics Department, Iowa State University Peimeng Yin Mathematics Department, Iowa State University

Analysis of PDEs Numerical Analysis and Scientific Computing mathscidoc:2203.03001

J Sci Comput, 77, 467-501, 2018.6
A novel discontinuous Galerkin (DG) method is developed to solve timedependent bi-harmonic type equations involving fourth derivatives in one and multiple space dimensions. We present the spatial DG discretization based on a mixed formulation and central interface numerical fluxes so that the resulting semi-discrete schemes are L^2 stable even without interior penalty. For time discretization, we use Crank–Nicolson so that the resulting scheme is unconditionally stable and second order in time. We present the optimal L^2 error estimate of O(h^{k+1}) for polynomials of degree k for semi-discrete DG schemes, and the L2 error of O(h^{k+1} + (\Delta t)^2) for fully discrete DG schemes. Extensions to more general fourth order partial differential equations and cases with non-homogeneous boundary conditions are provided. Numerical results are presented to verify the stability and accuracy of the schemes. Finally, an application to the one-dimensional Swift–Hohenberg equation endowed with a decay free energy is presented.
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@inproceedings{hailiang2018a,
  title={A Mixed Discontinuous Galerkin Method Without Interior Penalty for Time-Dependent Fourth Order Problems},
  author={Hailiang Liu, and Peimeng Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310094900343217910},
  booktitle={J Sci Comput},
  volume={77},
  pages={467-501},
  year={2018},
}
Hailiang Liu, and Peimeng Yin. A Mixed Discontinuous Galerkin Method Without Interior Penalty for Time-Dependent Fourth Order Problems. 2018. Vol. 77. In J Sci Comput. pp.467-501. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310094900343217910.
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