Abstract
An abstract property (H) is the key to a complete a priori error analysis in the (discrete) energy norm for several nonstandard finite element methods in the recent work [Lowest-order equivalent nonstandard finite element methods for biharmonic plates, Carstensen and Nataraj, M2AN, 2022]. This paper investigates the impact of (H) to the a posteriori error analysis and establishes known and novel explicit residual-based a posteriori error estimates. The abstract framework applies to Morley, two versions of discontinuous Galerkin, C0 interior penalty, as well as weakly over-penalized symmetric interior penalty schemes for the biharmonic equation with a general source term in H−2 (Ω).
Funding statement: The research of the first two authors have been supported by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 under the project ‘Foundation and Application of Generalized Mixed FEM Towards Nonlinear Problems in Solid Mechanics’ (CA 151/22-2). This paper has been supported by SPARC project (id 235) ‘The Mathematics and Computation of Plates’ and SERB POWER Fellowship SPF/2020/000019. The second author is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689).
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A A posteriori error control of a piecewise polynomial source in H–2(Ω)
This appendix provides an alternative view on the reliable and efficient estimator from Section 7 as lower and upper bounds for the dual norm of a piecewise polynomial source in H−2 (Ω). Suppose the piecewise polynomials Λ0 ∈ Pk (T), Λ1 ∈ Pk (T;R2), and Λ2 ∈ Pk (T; S) define the linear functional Λ ∈ H−2 (Ω) by
Recall the transfer operators IM, Ih , Jh for the five quadratic discretization schemes of Section 4 listed in Table 1. A reliable and efficient estimator μ2 (T) :=
Theorem A.1 (reliability and efficiency). There exist positive constants Crel , Ceff > 0 that exclusively depend on the shape regularity of T and on the polynomial degree k ∈ N0 such that
Proof. The discussion in Subsection 7.2 applies to Res := F := Λ and uh := 0 with apx(F, T) = 0. In this particular case, Proposition 7.1 provides the first inequality
with Crel = C4. Let u ∈
follows from Proposition 7.2 with Ceff = C8.
Theorem A.1 allows for a direct application to the linearization of semilinear problems in [18]. It can be further generalized in various directions, e.g., in the spirit of Section 7 that considers the a posteriori error analysis of the linear biharmonic problem for a more general class of functionals in H−2 (Ω) including line and point loads. The reliability requires only piecewise smoothness of Λ0, Λ1, Λ2 so that the traces and derivatives in μ1, μ2, μ3 exist, while the efficiency may require extra oscillation terms (as in (7.5)).
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