C 0-hybrid high-order methods for biharmonic problems

Abstract

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