Discrete p-robust $\varvec{H}({{\mathrm{div}}})$-liftings and a posteriori estimates for elliptic problems with $H^{-1}$ source terms

Abstract

We establish the existence of liftings into discrete subspaces of $\varvec{H}({{\mathrm{div}}})$ of piecewise polynomial data on locally refined simplicial partitions of polygonal/polyhedral domains. Our liftings are robust with respect to the polynomial degree. This result has important applications in the a posteriori error analysis of parabolic problems, where it permits the removal of so-called transition conditions that link two consecutive meshes. It can also be used in the a posteriori error analysis of elliptic problems, where it allows the treatment of meshes with arbitrary numbers of hanging nodes between elements. We present a constructive proof based on the a posteriori error analysis of an auxiliary elliptic problem with $H^{-1}$ source terms, thereby yielding results of independent interest. In particular, for such problems, we obtain guaranteed upper bounds on the error along with polynomial-degree robust local efficiency of the estimators.

Application to a posteriori error estimates on meshes with hanging nodes

Equilibrated flux a posteriori error estimates for meshes with hanging nodes are developed in [11] where the equilibration is performed on patches $\omega _{\mathbf {a}}$ corresponding to the support of hat functions $\psi _{\mathbf {a}}$ associated with non-hanging nodes $\mathbf {a}$ of the computational mesh and forming a partition of unity of the computational domain, see [11, Assumption 2.1]. It turns out that a slight extension of the equilibration patch $\omega _{\mathbf {a}}$ of [11] enables the removal of the usual dependence of a posteriori efficiency constants on the number of levels of hanging nodes, thereby allowing for a completely arbitrary number of levels of hanging nodes. More precisely, it suffices to extend the equilibration patch $\omega _{\mathbf {a}}$ so that all the products $h_{\omega _{\mathbf {a}}} ||\nabla \psi _{\mathbf {a}}||_{\infty , \omega _{\mathbf {a}}}$ are uniformly bounded. Then, applying Theorem 1.2 where the patch $\omega _{\mathbf {a}}$ of [11] corresponds here to the domain $\varOmega$, and the hat function $\psi _{\mathbf {a}}$ of [11] corresponds here to the function $\psi _{\dagger }$, we infer that the factor $h_{\omega _{\mathbf {a}}} \max _{\hat{\mathbf {a}} \in \widehat{\mathcal {V}}_{\mathbf {a}}} ||\nabla \psi _{\hat{\mathbf {a}}}||_{\infty ,\omega _{\hat{\mathbf {a}}}}$ of [11, Theorem 3.12] can be replaced by the factor $h_{\omega _{\mathbf {a}}} ||\nabla \psi _{\mathbf {a}}||_{\infty , \omega _{\mathbf {a}}}$, i.e. $h_{\varOmega }||\nabla \psi _{\dagger }||_{\infty }$ in the present notation. The extension of the equilibration patch is illustrated in Fig. 1. This extension typically entails including several layers of fine elements, so as to ensure that the factors $h_{\omega _{\mathbf {a}}} ||\nabla \psi _{\mathbf {a}}||_{\infty , \omega _{\mathbf {a}}}$ are uniformly bounded. The price to pay to achieve robustness with respect to the level of hanging nodes is thus a somewhat more expensive computation of the equilibrated flux. The proof of Theorem 1.2 in Sect. 6 shows that this cost can be significantly reduced to the solution of two low-order systems over the extended patch, followed by local high-order corrections within the extended patch.

Fig. 1

Original equilibration patches of Ref. [11] (left) and extended equilibration patches necessary for estimates robust with respect to an arbitrary number of levels of hanging nodes (right)