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We devise and analyze two hybrid high-order (HHO) methods for the numericalapproximation of the biharmonic problem. The methods support polyhedral meshes,rely on the primal formulation of the problem, and deliver $O(h^{k+1})$$H^2$-error estimates when using polynomials of order $k\ge0$ to approximatethe normal derivative on the mesh (inter)faces. Both HHO methods hinge on astabilization in the spirit of Lehrenfeld--Sch\"oberl for second-order PDEs.The cell unknowns are polynomials of order $(k+2)$ that can be eliminatedlocally by means of static condensation. The face unknowns approximating thetrace of the solution on the mesh (inter)faces are polynomials of order $(k+1)$in the first HHO method which is valid in dimension two and uses an originalstabilization involving the canonical hybrid finite element, and they are oforder $(k+2)$ for the second HHO method which is valid in arbitrary dimensionand uses only $L^2$-orthogonal projections in the stabilization. A comparativediscussion with the weak Galerkin methods from the literature is provided,highlighting the close connections and the improvements proposed herein.Additionally, we show how the two HHO methods can be combined with aNitsche-like boundary-penalty technique to weakly enforce the boundaryconditions. An originality in the devised Nitsche's technique is to avoid anypenalty parameter that must be large enough. Finally, numerical resultsshowcase the efficiency of the proposed methods and indicate that the HHOmethods can generally outperform discontinuous Galerkin methods and even becompetitive with $C^0$-interior penalty methods on triangular meshes.

Hybrid high-order and weak Galerkin methods for the biharmonic problem