An adaptive $hp$-refinement strategy with computable guaranteed bound on the error reduction factor

Abstract: We propose a new practical adaptive refinement strategy for $hp$-finite element approximations of elliptic problems. Following recent theoretical developments in polynomial-degree-robust a posteriori error analysis, we solve two types of discrete local problems on vertex-based patches. The first type involves the solution on each patch of a mixed finite element problem with homogeneous Neumann boundary conditions, which leads to an ${\mathbf H}(\mathrm{div},\Omega)$-conforming equilibrated flux. This, in turn, yields a guaranteed upper bound on the error and serves to mark mesh vertices for refinement via D\"orfler's bulk-chasing criterion. The second type of local problems involves the solution, on patches associated with marked vertices only, of two separate primal finite element problems with homogeneous Dirichlet boundary conditions, which serve to decide between $h$-, $p$-, or $hp$-refinement. Altogether, we show that these ingredients lead to a computable guaranteed bound on the ratio of the errors between successive refinements (error reduction factor). In a series of numerical experiments featuring smooth and singular solutions, we study the performance of the proposed $hp$-adaptive strategy and observe exponential convergence rates. We also investigate the accuracy of our bound on the reduction factor by evaluating the ratio of the predicted reduction factor relative to the true error reduction, and we find that this ratio is in general quite close to the optimal value of one.