Effect of numerical integration for elliptic obstacle problems

Jiang, Xun; Nochetto, Ricardo H.

doi:10.1007/s002110050041

Cited by 11 publications

(6 citation statements)

References 15 publications

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“…We thus assume = h . We also assume that {T h } is a quasi-uniform family and that the family {T h } is weakly acute [11] in the sense that…”

Section: Assumptions and A Numerical Methodsmentioning

confidence: 99%

Global dynamics of approximate solutions to an age-structured epidemic model with diffusion

Kim

2006

Adv Comput Math

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We consider an age-dependent s-i-s epidemic model with diffusion whose mortality is unbounded. We approximate the solution using Galerkin methods in the space variable combined with backward Euler along the characteristic direction in the age and time variables. It is proven that the scheme is stable and convergent in optimal rate in l ∞,2 (L 2 ) norm. To investigate the global behavior of the discrete solution resulting from the algorithm, we reformulate the resulting system into a monotone form. Positivity of the nonlocal birth process is proved using the positivity of the first eigenvalue of the resulting matrix system and using the fact that the positivity is preserved along the characteristics. The difference equation of the steady state coupled with nonlocal birth process is solved by developing monotone iterative schemes. The stability of the discrete solution of the steady state is then analyzed by constructing suitable positive subsolutions.

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“…We thus assume = h . We also assume that {T h } is a quasi-uniform family and that the family {T h } is weakly acute [11] in the sense that…”

Section: Assumptions and A Numerical Methodsmentioning

confidence: 99%

Global dynamics of approximate solutions to an age-structured epidemic model with diffusion

Kim

2006

Adv Comput Math

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“…Let us now prove the statement (ii). We follow Jiang and Nochetto [257]. Assume that A is a Z-matrix.…”

Section: Proof the Statement (I) Follows From The Fact Thatmentioning

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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“…Finally, let us stress that we assume exact integration for the non-smooth nonlinearity β ε (u ε,h − ψ). We refer to [Jiang and Nochetto, 1994], where a lumping technique is used for the numerical approximation of the non-linear term.…”

Section: The Discrete Problemmentioning

confidence: 99%

Finite element error estimates in $L^2$ for regularized discrete approximations to the obstacle problem

Hafemeyer¹,

Kahle²,

Pfefferer³

2018

Preprint

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This work is concerned with quasi-optimal a-priori finite element error estimates for the obstacle problem in the L 2 -norm. The discrete approximations are introduced as solutions to a finite element discretization of an accordingly regularized problem. The underlying domain is only assumed to be convex and polygonally or polyhedrally bounded such that an application of pointwise error estimates results in a rate less than two in general. The main ingredient for proving the quasi-optimal estimates is the structural and commonly used assumption that the obstacle is inactive on the boundary of the domain. Then localization techniques are used to estimate the global L 2 -error by a local error in the inner part of the domain, where higher regularity for the solution can be assumed, and a global error for the Ritz-projection of the solution, which can be estimated by standard techniques. We validate our results by numerical examples.

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Effect of numerical integration for elliptic obstacle problems

Cited by 11 publications

References 15 publications

Global dynamics of approximate solutions to an age-structured epidemic model with diffusion

Global dynamics of approximate solutions to an age-structured epidemic model with diffusion

Finite Elements II

Finite element error estimates in $L^2$ for regularized discrete approximations to the obstacle problem

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