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A primal discontinuous Galerkin method with static condensation on very general meshes

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Abstract

We propose an efficient variant of a primal discontinuous Galerkin method with interior penalty for the second order elliptic equations on very general meshes (polytopes with eventually curved boundaries). Efficiency, especially when higher order polynomials are used, is achieved by static condensation, i.e. a local elimination of certain degrees of freedom cell by cell. This alters the original method in a way that preserves the optimal error estimates. Numerical experiments confirm that the solutions produced by the new method are indeed very close to that produced by the classical one.

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Notes

  1. The usual SIP DG method makes perfect sense also for piecewise linear polynomials (\(k=1\)). We restrict ourselves however to \(k\ge 2\) since the forthcoming modification of the method allowing for the static condensation is pertinent to this case only.

  2. The usual choice \(h_E={{\,\mathrm{diam}\,}}(E)\) is not appropriate on general meshes since some of the facets can be of much smaller diameter than that of the adjacent cell.

  3. More precisely, all solutions \(u_h\) of (14)–(18) may be accompanied by \(p_h\in M_h\) so that the resulting couples \((u_h,p_h)\) also solve (30)–(31). Since the solution to (30)–(31) is unique, the inverse statement is also true: \(u_h\) given by (30)–(31) is also a solution to (14)–(18).

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Acknowledgements

I am grateful to Simon Lemaire for interesting discussions about HHO and msHHO methods, which were the starting point of conceiving the present article. I also thank the anonymous reviewers for very careful reading of the manuscript and for helpful remarks and suggestions.

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  1. Laboratoire de Mathématiques de Besançon, CNRS UMR 6623, Univ. Bourgogne Franche-Comté, 16 route de Gray, 25030, Besançon Cedex, France

    Alexei Lozinski

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Lozinski, A. A primal discontinuous Galerkin method with static condensation on very general meshes. Numer. Math. 143, 583–604 (2019). https://doi.org/10.1007/s00211-019-01067-1

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