Abstract
In most finite element methods the mesh is used to both represent the domain and to define the finite element basis. As a result the quality of such methods is tied to the quality of the mesh and may suffer when the latter deteriorates. This paper formulates an alternative approach, which separates the discretization of the domain, i.e., the meshing, from the discretization of the PDE. The latter is accomplished by extending the Generalized Moving Least-Squares (GMLS) regression technique to approximation of bilinear forms and using the mesh only for the integration of the GMLS polynomial basis. Our approach yields a non-conforming discretization of the weak equations that can be handled by standard discontinuous Galerkin or interior penalty terms.
Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525.
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We recall that \(\varPhi \)-unisolvency implies \(\{\phi \in \varPhi \,|\, \lambda _i(\phi ) = 0, i=1,\ldots ,n\} = \{0\}\).
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Acknowledgments
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research under Award Number DE-SC-0000230927, and the Laboratory Directed Research and Development program at Sandia National Laboratories.
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Bochev, P., Trask, N., Kuberry, P., Perego, M. (2020). Mesh-Hardened Finite Element Analysis Through a Generalized Moving Least-Squares Approximation of Variational Problems. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2019. Lecture Notes in Computer Science(), vol 11958. Springer, Cham. https://doi.org/10.1007/978-3-030-41032-2_7
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DOI: https://doi.org/10.1007/978-3-030-41032-2_7
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