A quasi-optimal variant of the Hybrid High-Order method for elliptic PDEs with $H^{-1}$ loads
Provides a quasi-optimal variant for a broader class of loads, addressing a theoretical gap for practitioners using Hybrid High-Order methods.
The paper extends Hybrid High-Order methods for elliptic PDEs to handle loads in $H^{-1}(Ω)$, achieving $H^1$-norm error bounded solely by the best approximation error in broken polynomial spaces, with improved $L^2$-norm error via duality.
Hybrid High-Order methods for elliptic diffusion problems have been originally formulated for loads in the Lebesgue space $L^2(Ω)$. In this paper we devise and analyze a variant thereof, which is defined for any load in the dual Sobolev space $H^{-1}(Ω)$. 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 Hybrid High-Order 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 $H^1_0(Ω)$.