$hp$-a posteriori error estimates for hybrid high-order methods applied to biharmonic problems

We derive a residual-based $hp$-a posteriori error estimator for hybrid high-order (HHO) methods on simplicial meshes applied to the biharmonic problem posed on two- and three-dimensional polytopal Lipschitz domains. The a posteriori error estimator hinges on an error decomposition into conforming and nonconforming components. To bound the nonconforming error, we use a $C^1$-partition of unity constructed via Alfeld splittings, combined with local Helmholtz decompositions on vertex stars where the key contribution is to show that the stability constant only depends on the mesh shape-regularity. For the conforming error, we design two residual-based estimators, each associated with a specific interpolation operator. In the first setting, the upper bound on the conforming error involves only the stabilization term and the data oscillation, but hinges on an assumption that we verify numerically. In the second setting, the bound additionally incorporates bulk residuals, normal flux jumps, and tangential jumps. Numerical experiments confirm the theoretical findings on the error upper bound and also illustrate numerically that the proposed estimators lead to moderate effectivity indices.