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 which 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 $k\ge0$ 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 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 (IPDG) methods as well. The present technique does not require bubble functions or a $C^1$-smoother to evaluate the right-hand side in case of rough loads. Finally, numerical results including comparisons to various existing methods showcase the efficiency of the proposed $C^0$-conforming HHO methods.

翻译：我们设计并分析了$C^0$连续的混合高阶方法，用于逼近具有固支或简支边界条件的双调和问题。$C^0$连续的混合高阶方法依赖于单元未知量和面未知量：单元未知量为网格单元内逼近解的$(k+2)$阶$C^0$连续多项式，面未知量为网格骨架上逼近解的法向导数的$k\ge0$阶多项式。此类方法对于光滑解可达到$O(h^{k+1})$的$H^2$误差估计。误差分析中的一个重要创新是降低了对精确解的最低正则性要求。实现这一目标的技术不仅适用于$C^0$连续的混合高阶方法，为说明此点，我们还概述了著名的$C^0$连续内部惩罚间断伽辽金方法的误差分析。该技术无需使用气泡函数或$C^1$光滑化算子来处理粗糙载荷情况下的右端项。最后，与多种现有方法比较的数值结果展示了所提出的$C^0$连续混合高阶方法的有效性。