Hybrid High-Order and Weak Galerkin Methods for the Biharmonic Problem

We devise and analyze two hybrid high-order (HHO) methods for the numerical approximation of biharmonic problem. The support polyhedral meshes, rely on primal formulation problem, deliver $O(h^{k+1})$ $H^2$-error estimates when using polynomials order $k\ge0$ to approximate normal derivative mesh (inter)faces. Both HHO hinge a stabilization in spirit Lehrenfeld Schöberl second-order PDEs. cell unknowns are $(k+2)$ that can be eliminated locally by means static condensation. face approximating trace solution (inter)faces $(k+1)$ first method, which is valid dimension uses an original involving canonical finite element, they second arbitrary only $L^2$-orthogonal projections stabilization. A comparative discussion with weak Galerkin from literature provided, highlighting close connections improvements proposed herein. Additionally, we show how combined Nitsche-like boundary-penalty technique weakly enforce boundary conditions. An originality devised Nitsche's avoid any penalty parameter must large enough. Finally, results showcase efficiency indicate generally outperform discontinuous even competitive $C^0$-interior triangular meshes.