Abstract
Abstract For the Poisson problem in two dimensions, posed on a domain partitioned into axis-aligned rectangles with up to one hanging node per edge, we envision an efficient error reduction step in an instance-optimal hp-adaptive finite element method. Central to this is the problem: Which increase in local polynomial degree ensures p-robust contraction of the error in energy norm? We reduce this problem to a small number of saturation problems on the reference square, and provide strong numerical evidence for their solution.
Highlights
We consider the Poisson model problem of finding u : Ω → R that satisfies (1.1)−△u = f in Ω, u = 0 on ∂Ω, where Ω ⊂ R2 is a connected union of a finite number of essentially disjoint axis-aligned rectangles, and f ∈ L2(Ω)
T that vanish on the domain boundary, and let uT ∈ UT be its best approximation of u in energy norm
It is well known that this problem is equivalent to the saturation problem of finding T for which ∇uT − ∇uT L2(Ω) ≤ ρ ∇u − ∇uT L2(Ω) for some ρ > 1; in this work, we will study the saturation problem, posed locally on a patch of rectangles around a given vertex
Summary
In [5], the error reducer of step (i) was a typical h-adaptive loop driven by an element-based Dorfler marking, using the a posteriori error estimator of Melenk and Wohlmuth [13] The efficiency of this error estimator is known to be sensitive to polynomial degrees, which can lead to a runtime that grows exponentially in the number of DoFs. In [6], Canuto et al explore a different error reduction strategy. It is an adaptive p-enrichment loop driven by a vertex-based Dorfler marking using the equilibrated flux estimator, which was shown to be p-robust in [2] They show that solving a number of local saturation problems, posed on patches around a vertex in terms of dual norms of residuals, leads to an efficient error reducer.
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