An improved a priori error analysis of Nitsche’s method for Robin boundary conditions

Lüthen, Nora; Juntunen, Mika; Stenberg, Rolf

doi:10.1007/s00211-017-0927-1

Cited by 14 publications

(15 citation statements)

References 8 publications

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“…It results that Γ − C can be viewed as a discrete approximation of the actual contact surface. On Γ + C a free Neumann boundary condition is imposed weakly, in the same fashion as in [62,73]. Therefore Γ + C may represent a discrete approximation of the unsticked contact surface.…”

Section: Nitsche For Contact and Nitsche For Dirichletmentioning

confidence: 99%

An Overview of Recent Results on Nitsche’s Method for Contact Problems

Chouly

Fabre

Hild

et al. 2017

Lecture Notes in Computational Science and Engineering

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We summarize recent achievements in applying Nitsche's method to some contact and friction problems. We recall the setting of Nitsche's method in the case of unilateral contact with Tresca friction in linear elasticity. Main results of the numerical analysis are detailed: consistency, well-posedness, fully optimal convergence in H 1 (Ω)-norm, residual-based a posteriori error estimation. Some numerics and some recent extensions to multibody contact, contact in large transformations and contact in elastodynamics are presented as well.

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Section: Nitsche For Contact and Nitsche For Dirichletmentioning

confidence: 99%

An Overview of Recent Results on Nitsche’s Method for Contact Problems

Chouly

Fabre

Hild

et al. 2017

Lecture Notes in Computational Science and Engineering

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“…Remark 41.10 (Literature). An alternative analysis based on the approach of Gudi [226] is developed in Lüthen et al [291].…”

Section: Nitsche's Boundary Penalty Methodsmentioning

confidence: 99%

Finite Elements II

Ern¹,

Guermond²

2021

Texts in Applied Mathematics

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Exercise . The goal of this exercise is to derive another proof of the Lax-Milgram lemma. Let A ∈ L(V ; V ) be defined by (A(v), w) V := a(v, w) for all v, w ∈ V (note that we use an inner product to define A). Let L be the representative in V of the linear form ℓ ∈ V ′ . Let λ be a positive real number. Consider the mapwith ρ λ ∈ (0, 1), and show that (25.6) is well-posed. (Hint : use Banach's fixed-point theorem.) Exercise 25.7 (Coercivity as necessary condition). Let V be a reflexive Banach space and let A ∈ L(V ; V ′ ) be a monotone self-adjoint operator; see Definition C.31. Prove that A is bijective if and only if A is coercive (with ξ := 1). (Hint : prove that ℜ(for all v, w ∈ V.) Exercise 25.8 (Darcy). Prove that the problem (24.14) is well-posed. (Hint : adapt the proof of Proposition 25.18.) Exercise 25.9 (First-order PDE). Prove that the problem (24.21) is well-posed. (Hint : adapt the proof of Proposition 25.19.) Exercise 25.10 (T -coercivity). Let V, W be Hilbert spaces. Prove that (bnb1)-(bnb2) are equivalent to the existence of a bijective operator T ∈ L(V ; W ) and a real number η > 0 such that ℜ(a(v, T (v))) ≥ η v 2V for all v ∈ V. (Hint : use J −1 W , (A −1 ) * , and the map J rf V from the Riesz-Fréchet theorem to construct T .) Exercise 25.11 (Sign-changing diffusion). Let D be a Lipschitz domain D in R d partitioned into two disjoint Lipschitz subdomains D 1 and D 2 . Set Σ := ∂D 1 ∩∂D 2 , each having an intersection with ∂D of positive measure. Let κ 1 , κ 2 be two real numbers s.t. κ 1 > 0 and κ 2 < 0. Set κ(x) := κ 1 1 D1 (x)+κ 2 1 D2 (x) for all x ∈ D. Let V := H 1 0 (D) be equipped with the norm ∇v L 2 (D) . The goal is to show that the bilinear form a(v, w) := D κ∇v•∇w satisfies conditions (bnb1)-(bnb2) on V ×V ; see Chesnel and Ciarlet [118]. Set V m := {v| Dm | v ∈ V } for all m ∈ {1, 2}, equipped with the norm ∇v m L 2 (Dm) for all v m ∈ V m , and let γ 0,m be the traces of functions in V m on Σ. (i) Assume that there isProve that T ∈ L(V ) and that T is an isomorphism. (Hint : verify that T • T = I V , the identity in V.) (ii) Assume that κ1 |κ2| > S 1 2 L(V1;V2) . Prove that the conditions (bnb1)-(bnb2) are satisfied.

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“…In [21], Nitsche introduced a consistent penalty-type method for imposing Dirichlet boundary conditions in the second-order Poisson problem. Nitsche's method was extended to other boundary conditions (in particular, inhomogeneous Robin) in Juntunen and Stenberg [16] by unifying the implementation and analysis via a parameterdependent boundary value problem; an improved a priori analysis was presented in Lüthen, Juntunen, and Stenberg [19]. Different boundary conditions (Dirichlet, Neumann, Robin) were obtained by changing the value of a single nonnegative parameter.…”

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confidence: 99%

“…In this study we explore the above ideas [16,19,21] in the context of fourthorder H 2 -conforming problems. In particular, we seek to unify the implementation and the analysis of different boundary conditions for the Kirchhoff-Love plate equation [17,18] by presenting Nitsche's method, which incorporates the boundary conditions in the discrete formulation as consistent penalty terms.…”

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confidence: 99%