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Journal Articles IMA Journal of Numerical Analysis Year : 2021

Bridging the Hybrid High-Order and Virtual Element methods

Abstract

We present a unifying viewpoint at Hybrid High-Order and Virtual Element methods on general polytopal meshes in dimension 2 or 3, both in terms of formulation and analysis. We focus on a model Poisson problem. To build our bridge, (i) we transcribe the (conforming) Virtual Element method into the Hybrid High-Order framework, and (ii) we prove $H^m$ approximation properties for the local polynomial projector in terms of which the local Virtual Element discrete bilinear form is defined. This allows us to perform a unified analysis of Virtual Element/Hybrid High-Order methods, that differs from standard Virtual Element analyses by the fact that the approximation properties of the underlying virtual space are not explicitly used. As a complement to our unified analysis, we also study interpolation in local virtual spaces, shedding light on the differences between the conforming and nonconforming cases.
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Dates and versions

hal-01902962 , version 1 (24-10-2018)
hal-01902962 , version 2 (11-11-2018)
hal-01902962 , version 3 (04-09-2019)
hal-01902962 , version 4 (09-09-2019)
hal-01902962 , version 5 (17-10-2019)

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Cite

Simon Lemaire. Bridging the Hybrid High-Order and Virtual Element methods. IMA Journal of Numerical Analysis, 2021, 41 (1), pp.549-593. ⟨10.1093/imanum/drz056⟩. ⟨hal-01902962v5⟩
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