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

摘要

We propose a new practical adaptive refinement strategy for $hp$-finiteelement approximations of elliptic problems. Following recent theoreticaldevelopments in polynomial-degree-robust a posteriori error analysis, we solvetwo types of discrete local problems on vertex-based patches. The first typeinvolves the solution on each patch of a mixed finite element problem withhomogeneous Neumann boundary conditions, which leads to an ${\mathbfH}(\mathrm{div},\Omega)$-conforming equilibrated flux. This, in turn, yields aguaranteed upper bound on the error and serves to mark mesh vertices forrefinement via D\"orfler's bulk-chasing criterion. The second type of localproblems involves the solution, on patches associated with marked verticesonly, of two separate primal finite element problems with homogeneous Dirichletboundary conditions, which serve to decide between $h$-, $p$-, or$hp$-refinement. Altogether, we show that these ingredients lead to acomputable guaranteed bound on the ratio of the errors between successiverefinements (error reduction factor). In a series of numerical experimentsfeaturing smooth and singular solutions, we study the performance of theproposed $hp$-adaptive strategy and observe exponential convergence rates. Wealso investigate the accuracy of our bound on the reduction factor byevaluating the ratio of the predicted reduction factor relative to the trueerror reduction, and we find that this ratio is in general quite close to theoptimal value of one.