Localization of the W-1,q norm for local a posteriori efficiency

Abstract

This paper gives a direct proof of localization of dual norms of bounded linear functionals on the Sobolev space ${W^{1,p}_0(\varOmega )}$, $1 \leq p \leq \infty $. The basic condition is that the functional in question vanishes over locally supported test functions from ${W^{1,p}_0(\varOmega )}$ which form a partition of unity in $\varOmega $, apart from close to the boundary $\partial \varOmega $. We also study how to weaken this condition. The results allow in particular to establish local efficiency and robustness with respect to the exponent $p$ of a posteriori estimates for nonlinear partial differential equations in divergence form, including the case of inexact solvers. Numerical illustrations support the theory.

Guaranteed, Locally Space-Time Efficient, and Polynomial-Degree Robust a Posteriori Error Estimates for High-Order Discretizations of Parabolic Problems

“…Proof of (5.14) and The proof is essentially a counting argument after local Riesz mappings are introduced to evaluate the negative norms ∂ t (u − Iu hτ ) 2 H −1 (ωa) for all a ∈ V n , see also [2]. Summing (5.12) over K ∈ T n and using (8.10) then leads to K∈T…”

Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients

“…on the intrinsic error u − u h * given by (2.2) seem rather Σand T-dependent, see in particular the numerical study in Section 6 of [22]. Finally, Figure 5 illustrates that the distribution of the error is predicted very correctly by our estimators (plotting by a piecewise affine function is done as explained in Section 5 of [8]).…”

“…Remark 3.4 (Further generalization). Theorem 3.3 has recently been extended to any bounded linear functional on the Sobolev space W 1,p 0 (Ω), p > 1, in [8].…”

“…We will focus on our a posteriori error estimates of Theorem 4.5, while tracing the error u − u h * defined by (2.1a) In what follows, we compute u − u h * approximately, while approximating (5.4). We again employ a posteriori error estimates to ensure that u − u h * is computed with relative accuracy 10 −2 , see the details in Section 5 of [8]. We will also display the canonical H 1 0 (Ω)-norm of the error ∇(u − u h ) .…”

“…For dual (residual) norms, a result of this type has probably first been shown in Babuška and Miller Theorem 2.1.1 of [4], and may be deduced from more recent a posteriori analyses, see in particular Carstensen and Funken [17], Morin et al [38], Verfürth [48,50], Veeser and Verfürth [46], Cohen et al [24], and the references therein, typically for piecewise polynomial approximations. It has recently been extended in [8] to any bounded linear functional on the Sobolev space W 1,p 0 (Ω), p > 1. Galerkin orthogonality with respect to lowest-order modes turns out to be crucial here for one direction of the equivalence.…”