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We devise and analyze $C^0$-conforming hybrid high-order (HHO) methods toapproximate biharmonic problems with either clamped or simply supportedboundary conditions. $C^0$-conforming HHO methods hinge on cell unknowns whichare $C^0$-conforming polynomials of order $(k+2)$ approximating the solution inthe mesh cells and on face unknowns which are polynomials of order $k\ge0$approximating the normal derivative of the solution on the mesh skeleton. Suchmethods deliver $O(h^{k+1})$ $H^2$-error estimates for smooth solutions. Animportant novelty in the error analysis is to lower the minimal regularityrequirement on the exact solution. The technique to achieve this has broaderapplicability than just $C^0$-conforming HHO methods, and to illustrate thispoint, we outline the error analysis for the well-known $C^0$-conforminginterior penalty discontinuous Galerkin (IPDG) methods as well. The presenttechnique does not require bubble functions or a $C^1$-smoother to evaluate theright-hand side in case of rough loads. Finally, numerical results includingcomparisons to various existing methods showcase the efficiency of the proposed$C^0$-conforming HHO methods.

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