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Journal Articles IMA Journal of Numerical Analysis Year : 2024

$C^0$-hybrid high-order methods for biharmonic problems

Abstract

We devise and analyze $C^0$-conforming hybrid high-order (HHO) methods to approximate biharmonic problems with either clamped or simply supported boundary conditions. $C^0$-conforming HHO methods hinge on cell unknowns which are $C^0$-conforming polynomials of order $(k+2)$ approximating the solution in the mesh cells and on face unknowns which are polynomials of order $k\ge0$ approximating the normal derivative of the solution on the mesh skeleton. Such methods deliver $O(h^{k+1})$ $H^2$-error estimates for smooth solutions. An important novelty in the error analysis is to lower the minimal regularity requirement on the exact solution. The technique to achieve this has a broader applicability than just $C^0$-conforming HHO methods, and to illustrate this point, we outline the error analysis for the well-known $C^0$-conforming interior penalty discontinuous Galerkin (IPDG) methods as well. The present technique does not require bubble functions or a $C^1$-smoother to evaluate the right-hand side in case of rough loads. Finally, numerical results including comparisons to various existing methods showcase the efficiency of the proposed $C^0$-conforming HHO methods.
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Dates and versions

hal-03695484 , version 1 (14-06-2022)
hal-03695484 , version 2 (07-01-2023)
hal-03695484 , version 3 (11-02-2023)

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Zhaonan Dong, Alexandre Ern. $C^0$-hybrid high-order methods for biharmonic problems. IMA Journal of Numerical Analysis, 2024, 44 (1), pp.24-57. ⟨10.1093/imanum/drad003⟩. ⟨hal-03695484v3⟩
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