This paper proposes a new hybrid high-order discretization for the biharmonic
problem and the corresponding eigenvalue problem. The discrete ansatz space
includes degrees of freedom in $n-2$ dimensional submanifolds (z.B., nodal
values in 2D and edge values in 3D), in addition to the typical degrees of
freedom in the mesh and on the hyperfaces in the HHO literature. This approach
enables the characteristic commuting property of the hybrid high-order
methodology in any space dimension and allows for lower eigenvalue bounds of
higher order for the eigenvalue problem. The main results are quasi-best
approximation estimates as well as reliable and efficient error control. Der
latter motivates an adaptive mesh-refining algorithm that empirically recovers
optimal convergence rates for singular solutions.

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