A quasi-optimal variant of the hybrid high-order method for elliptic partial differential equations with H−1 loads

Hybrid high-order (HHO) methods for elliptic diffusion problems have been originally formulated for loads in the Lebesgue space $L^2(\varOmega )$. In this paper we devise and analyse a variant thereof, which is defined for any load in the dual Sobolev space $H^{-1}(\varOmega )$. The main feature of the present variant is that its $H^1$-norm error can be bounded only in terms of the $H^1$-norm best error in a space of broken polynomials. We establish this estimate with the help of recent results on the quasi-optimality of nonconforming methods. We prove also an improved error bound in the $L^2$-norm by duality. Compared to previous works on quasi-optimal nonconforming methods the main novelties are that HHO methods handle pairs of unknowns and not a single function and, more crucially, that these methods employ a reconstruction that is one polynomial degree higher than the discrete unknowns. The proposed modification affects only the formulation of the discrete right-hand side. This is obtained by properly mapping discrete test functions into $H^1_0(\varOmega )$.