### UNIFORM STABILITY AND ERROR ANALYSIS FOR SOME DISCONTINUOUS GALERKIN METHODS

Qingguo Hong, Jinchao Xu

1. Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
• Received:2019-11-05 Revised:2020-03-09 Published:2021-03-15
• Contact: Jinchao Xu,Email:xu@math.psu.edu
• Supported by:
The authors would like to thank the referees for their constructive comments which improve the presentation of the paper. The work of both authors was partially supported by the Center for Computational Mathematics and Applications, The Pennsylvania State University, and was partially supported by NSF grant DMS-1522615.

Qingguo Hong, Jinchao Xu. UNIFORM STABILITY AND ERROR ANALYSIS FOR SOME DISCONTINUOUS GALERKIN METHODS[J]. Journal of Computational Mathematics, 2021, 39(2): 283-310.

In this paper, we provide a number of new estimates on the stability and convergence of both hybrid discontinuous Galerkin (HDG) and weak Galerkin (WG) methods. By using the standard Brezzi theory on mixed methods, we carefully define appropriate norms for the various discretization variables and then establish that the stability and error estimates hold uniformly with respect to stabilization and discretization parameters. As a result, by taking appropriate limit of the stabilization parameters, we show that the HDG method converges to a primal conforming method and the WG method converges to a mixed conforming method.

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