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

Abstract

We present a posteriori error analysis of diffusion problems where the diffusion tensor is not necessarily symmetric and positive definite and can in particular change its sign. We first identify the correct intrinsic error norm for such problems, covering both conforming and nonconforming approximations. It combines a dual (residual) norm together with the distance to the correct functional space. Importantly, we show the equivalence of both these quantities defined globally over the entire computational domain with the Hilbertian sums of their localizations over patches of elements. In this framework, we then design a posteriori estimators which deliver simultaneously guaranteed error upper bound, global and local error lower bounds, and robustness with respect to the (sign-changing) diffusion tensor. Robustness with respect to the approximation polynomial degree is achieved as well. The estimators are given in a unified setting covering at once conforming, nonconforming, mixed, and discontinuous Galerkin finite element discretizations in two or three space dimensions. Numerical results illustrate the theoretical developments.

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

“…In this case V a is reflexive, so for the sequence {r a j } ∞ j=1 from the proof of Theorem 3.7, there is a subsequence which converges weakly to some r a ∈ V a with ∇r a p,ωa ≤ lim inf j→∞ ∇r a j p,ωa = R q−1 (V a ) thanks to weak lower semicontinuity of norm and (3.24). On the other hand, from (3.25) and the weak convergence, we conclude that 26) which implies that R q−1 (V a ) ≤ ∇r a p,ωa . Hence, altogether we have that…”

“…See in particular [4, Theorem 2.1.1], Cohen et al [27, equation (3.23)], Ciarlet and Vohralík [26,Theorem 3.2], and the revised version of Ern and Guermond [31] for p = q = 2.…”

$hp$-Adaptation Driven by Polynomial-Degree-Robust A Posteriori Error Estimates for Elliptic Problems

“…The fourth problem involves a piecewise constant, anisotropic diffusion tensor and has no known exact solution. We extend our estimates to this case following [10]. We illustrate the asymptotic exponential order of convergence (5.1) of the hp-decision criteria PARAM for γ = 0.3, 0.6, PRIOR, and DECAY.…”

Mesh requirements for the finite element approximation of problems with sign-changing coefficients

“…• Let us mention that a posteriori error estimates for diffusion problems with sign-changing coefficients have been carried out in [14] and lead to similar meshing requirements near the interfaces.…”