Global superconvergence of finite element methods for parabolic inverse problems

Azari, Hossein; Zhang, Shuhua

doi:10.1007/s10492-009-0018-4

Cited by 3 publications

(1 citation statement)

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“…In particular, we can quote the contribution of Cannon and DuChateau in [4], where a time-dependent coefficient is identified in the space of Hölder-continuous function, under a linear (integral) overdeterminating condition, by means of fixed-point techniques. Also, in more recent papers [2,5,6], the existence of numerical solutions for a problem of this kind has been investigated, and convergence results have been established for time-approximation schemes. It is important to mention that, in all of these cases, the underlying domain is either a segment or a rectangle.…”

Section: Introductionmentioning

confidence: 99%

Recovering the reaction and the diffusion coefficients in a linear parabolic equation

Lorenzi¹,

Mola²

2012

Inverse Problems

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Let H be a real separable Hilbert space and A : D(A) → H be a positive and self-adjoint (unbounded) operator. We consider the identification problem consisting in searching for an H-valued function u and a couple of real numbers λ and μ, the first one being positive, that fulfil the initial-value problem and the additional constraintsA r/2 u(T ) 2 = ϕ and A s/2 u(T ) 2 = ψ, where we denote by A s and A r the powers of A with exponents r < s. Provided that the given data u 0 ∈ H, u 0 and ϕ, ψ > 0 satisfy proper a priori limitations, by means of a finite-dimensional approximation scheme, we construct a unique solution (u, λ, μ) on the whole interval [0, T ], and exhibit an explicit continuous dependence estimate of Lipschitz type with respect to the data. Also, we provide specific applications to second-and fourth-order parabolic initial-boundary-value problems.

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Section: Introductionmentioning

confidence: 99%