C⁰-hybrid high-order methods for biharmonic problems

We devise and analyze C⁰-conforming hybrid high-order (HHO) methods to approximate biharmonic problems with either clamped or simply supported boundary conditions. C⁰-conforming HHO methods hinge on cell unknowns that are C⁰-conforming polynomials of order (k + 2) approximating the solution in the mesh cells and on face unknowns, which are polynomials of order k ≥ 0 approximating the normal derivative of the solution on the mesh skeleton. Such methods deliver O(h^k+¹) H²-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 C⁰-conforming HHO methods, and to illustrate this point, we outline the error analysis for the well-known C⁰-conforming interior penalty discontinuous Galerkin methods as well. The present technique does not require a C¹smoother to evaluate the right-hand side in case of rough loads; loads in W^−1,q, q > _d²₊^d₂, are covered, but not in H⁻². Finally, numerical results including comparisons to various existing methods showcase the efficiency of the proposed C⁰-conforming HHO methods.