Error estimates and free-boundary convergence for a finite difference discretization of a parabolic variational inequality

Baiocchi, Claudio; Pozzi, Gianni A.

doi:10.1051/m2an/1977110403151

Cited by 17 publications

(7 citation statements)

References 15 publications

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“…x,~ D solution of the discrete problem as outlined above) some information may be available about the location of I due to a theorem by C. Baiocchi and G. A. Pozzi [3], see also [5]. We would like to have a more direct access to the information about I, the more so that in general (when u ~ C(D)) the variational technique provides no information about the location of the contact set.…”

Section: U(u) = Rain U(v)mentioning

confidence: 99%

Shape optimization approach to numerical solution of the Obstacle Problem

Bogomolny

Hou

1984

Appl Math Optim

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Abstract. We discuss an algorithm for the numerical solution of the Obstacle Problem in which the coincidence set is considered as the prime unknown. Domain functionals are defined for which the coincidence set serves as the minimizing element. Their gradients are computed (in the sense of the material derivative), and the gradient descent method employed to minimize these functionals. Numerical example is given.

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Section: U(u) = Rain U(v)mentioning

confidence: 99%

Shape optimization approach to numerical solution of the Obstacle Problem

Bogomolny

Hou

1984

Appl Math Optim

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“…Although uniqueness has been proved both in [20] and in [21], we report the following argument for its simplicity (see [29]). Theorem 3.1 (uniqueness).…”

Section: 3)mentioning

confidence: 99%

New results on some classical parabolic free-boundary problems

Fasano¹,

Primicerio²

1981

Quart. Appl. Math.

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With an appendix by A. LACEY (New College, Oxford) 1. Introduction. In [1] we analyzed a class of free-boundary problems for the heat equation in one space dimension, releasing the sign restrictions on the data and the latent heat usually required in the Stefan problem. Problems of this kind have been studied by other authors also in connection with the freezing of a supercooled liquid and with decision theory (see e.g. [2]- [16], and the references quoted herein).Two major problems remain open or not completely solved, namely (i) does any solution exist when the datum prescribed on the free boundary x = s(t) does not fit the initial datum at x = s(0)?; (ii) how are the data related to the possibility of continuing the solution in arbitrarily large time intervals?Sec. 2 of this paper contributes toward answering the above questions. Special results which are scattered in the literature cited can be found in the framework of our analysis (sometimes with relevant simplifications of the arguments).It is known that some free-boundary problems with the Cauchy data prescribed on the free boundary can be reduced to schemes of the type mentioned above, provided that suitable compatibility conditions are fulfilled by the data. A typical example in which such conditions are violated is given by the diffusion-consumption of oxygen in insulated living tissues, when the initial oxygen distribution coincides with the steady-state profile corresponding to a given constant input (see [14] and [17]-[23]).This case is considered in Sec. 3, there we prove the existence of a smooth solution and remark that the associated problem for the time derivative of the oxygen concentration is of the type considered in Sec. 2 but with an initial datum behaving like a "5-function " at the origin. A very sharp estimate of the lifetime of the tissue is also obtained by means of elementary calculations.In Sec. 4 we prove some comparison theorems for the solutions of the problems dealt with in the preceding sections.As an interesting consequence, a nonexistence theorem will identify a class of initial data such that the answer to question (i) above is negative.

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“…Baiocchi and Pozzi [2] gave an estimate for the location of the free boundary of the continuous solution using a shifting method. This technique relies on an L ∞ -error estimate of the discrete solution.…”

Section: Introductionmentioning

confidence: 99%