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Stable Discontinuous Galerkin FEM Without Penalty Parameters

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Numerical Mathematics and Advanced Applications ENUMATH 2015

Abstract

We propose a modified local discontinuous Galerkin (LDG) method for second–order elliptic problems that does not require extrinsic penalization to ensure stability. Stability is instead achieved by showing a discrete Poincaré–Friedrichs inequality for the discrete gradient that employs a lifting of the jumps with one polynomial degree higher than the scalar approximation space. Our analysis covers rather general simplicial meshes with the possibility of hanging nodes.

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Acknowledgements

The work of the third author was partially supported by the NSF grant DMS–1417980 and the Alfred Sloan Foundation.

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Correspondence to Iain Smears .

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John, L., Neilan, M., Smears, I. (2016). Stable Discontinuous Galerkin FEM Without Penalty Parameters. In: Karasözen, B., Manguoğlu, M., Tezer-Sezgin, M., Göktepe, S., Uğur, Ö. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2015. Lecture Notes in Computational Science and Engineering, vol 112. Springer, Cham. https://doi.org/10.1007/978-3-319-39929-4_17

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