A Galerkin method for mixed parabolic–elliptic partial differential equations

Stiemer, Marcus

doi:10.1007/s00211-010-0308-5

Cited by 8 publications

(6 citation statements)

References 21 publications

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“…It should be mentioned that the forward problem / is well‐posed in the sense of Hadamard. More precisely speaking, for any

q \in scriptA

, there exists a unique weak solution u ( x , t ), which satisfies the following regularity property (see Stiemer and Jiang et al)

u \in L^{\infty} false(false(0, T false); L^{2} false(ω false) false) \cap L^{2} false(false(0, T false); H_{0}^{1} false(normalΩ false) \cap H^{2} false(ω false) false), u_{t} \in L^{2} false(false(0, T false); L^{2} false(normalΩ false) false) .

…”

Section: Optimal Control Problemmentioning

confidence: 99%

“…In this work, we investigate an inverse problem of determining the diffusion coefficient in a parabolic‐elliptic system from the final overspecified data. Problems of this type arise in a large field of engineering and industrial applications (see, eg, previous studies and references therein). One important application is from electromagnetic metal forming (see other studies).…”

Section: Introductionmentioning

confidence: 99%

“…Problems of this type arise in a large field of engineering and industrial applications (see, eg, previous studies and references therein). One important application is from electromagnetic metal forming (see other studies). In fact, in the process of electromagnetic metal forming, the evolution of the deformation field of a mechanical structure consisting of well‐conducting material is coupled with an electromagnetic field, triggering a Lorentz force.…”

Section: Introductionmentioning

confidence: 99%

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An inverse problem of identifying the diffusion coefficient in a coupled parabolic‐elliptic system

Deng

Cai

Yang

2018

Math Methods in App Sciences

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This work studies a nonlinear inverse problem of reconstructing the diffusion coefficient in a parabolic-elliptic system using the final measurement data, which has important application in a large field of applied science. Being different from other works, which are governed by single partial differential equations, the underlying mathematical model in this paper is a coupled parabolic-elliptic system, which makes theoretical analysis rather difficult. On the basis of the optimal control framework, the identification problem is transformed into an optimization problem. Then the existence of the minimizer is proved, and the necessary condition that must be satisfied by the minimizer is also given. Since the optimal control problem is nonconvex, one may not expect a unique solution universally. However, the local uniqueness and stability of the minimizer are deduced in this paper. DENG ET AL 3415 coefficient problem of identification of the diffusion coefficient from the final overspecified data has been considered by several authors (see, for instance, Demir and Hasanov 22,23 ).Compared with other papers, which also deal with parameter identification problems, the results to be given in this paper have the following features. Firstly, the mathematical model proposed in this paper is a coupled parabolic-elliptic system rather than a single parabolic or elliptic equation. So this kind of problem indeed belongs to the boundary value problems of mixed type. Secondly, our task is to identify the diffusion coefficient in the coupled system, and this inverse problem generally belongs to the mathematical class of severely ill-posed problems. Furthermore, though the underlying mathematical model is linear, the corresponding inverse problem is completely nonlinear. Thirdly, since the exchange of matter and energy occurs on the interface, and after all, the connective condition is concerned with the diffusivity, the underlying mathematical model can be viewed as a nonlinear boundary-value problem combined with a nonlinear parameter-identification problem. From this point of view, such problem is rather difficult than those of single equation. Finally, the extra condition used in the paper is only imposed on the solution in the small domain rather than on the whole domain Ω. Because of the lack of measurement data, we shall carefully treat every integral to avoid using more information outside . In summary, the complexity of the inverse problem 1.2 to be introduced in following Section 3 is higher than those that are governed by single equations.In this work, we use an optimal control framework (see, eg, recent works 7,21,24,25 ) to discuss Problem 1.1 mainly from the theoretical analysis angle. As mentioned above, since the mathematical model in this paper is a coupled parabolic-elliptic system rather than the single parabolic equation in previous studies, 7,21,24,25 the major differences between them should be noted. It is well known that the properties of the solution for elliptic or parabolic system are quite different f...

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“…It should be mentioned that the forward problem / is well‐posed in the sense of Hadamard. More precisely speaking, for any

q \in scriptA

, there exists a unique weak solution u ( x , t ), which satisfies the following regularity property (see Stiemer and Jiang et al)

u \in L^{\infty} false(false(0, T false); L^{2} false(ω false) false) \cap L^{2} false(false(0, T false); H_{0}^{1} false(normalΩ false) \cap H^{2} false(ω false) false), u_{t} \in L^{2} false(false(0, T false); L^{2} false(normalΩ false) false) .

…”

Section: Optimal Control Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An inverse problem of identifying the diffusion coefficient in a coupled parabolic‐elliptic system

Deng

Cai

Yang

2018

Math Methods in App Sciences

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“…As the convection coefficient is positive in G + a boundary layer of width

O (ε)

appears on the left side of the boundary of G + , whereas the equation in (1.1) is of parabolic reaction‐diffusion type in G − which leads to occurrence of parabolic boundary layers of width

O (\sqrt{ε})

on both the left and right boundaries of G − and hence, the IBVP (1.1)–(1.3) in general possesses a boundary layer at x = 0 and interior layers of different widths in the neighborhood of the point

x = ξ

. These types of problems arise in the context of electromagnetic metal forming (see, e.g., ).…”

Section: Introductionmentioning

confidence: 99%

Uniform convergence analysis of hybrid numerical scheme for singularly perturbed problems of mixed type

Mukherjee

Natesan

2014

Numerical Methods Partial

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In this article, we consider a class of singularly perturbed mixed parabolic‐elliptic problems whose solutions possess both boundary and interior layers. To solve these problems, a hybrid numerical scheme is proposed and it is constituted on a special rectangular mesh which consists of a layer resolving piecewise‐uniform Shishkin mesh in the spatial direction and a uniform mesh in the temporal direction. The domain under consideration is partitioned into two subdomains. For the spatial discretization, the proposed scheme is comprised of the classical central difference scheme in the first subdomain and a hybrid finite difference scheme in the second subdomain, whereas the time derivative in the given problem is discretized by the backward‐Euler method. We prove that the method converges uniformly with respect to the perturbation parameter with almost second‐order spatial accuracy in the discrete supremum norm. Numerical results are finally presented to validate the theoretical results.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1931–1960, 2014

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“…Function u(x, t ) represents the profile of some physical quantity at time t and location x. The coupled parabolic-elliptic system (1.1) arises in many engineering and industrial applications; see [14][15][16][17][18] and references therein. One important application is from electromagnetic metal forming [16][17][18], where the evolution of the deformation field of a mechanical structure of some conducting material is coupled with an electromagnetic field that generates a Lorentz force, thus driving the metal forming process.…”

Section: Introductionmentioning

confidence: 99%

Convergence rates of Tikhonov regularizations for parameter identification in a parabolic-elliptic system

2012

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We shall study the convergence rates of the Tikhonov regularizations for the identification of the diffusivity q(x) in a parabolic–elliptic system. The H1 regularization and a mixed Lp–H1 regularization are considered. For the H1 regularization, we present a simple and easily interpretable source condition, under which the regularized solutions will be shown to converge at the standard rate in terms of the noise level of the data. The convergence is analyzed in three different approaches, which result in the same convergence rate but require quite different conditions on the measurement time and the identifying parameters. For the mixed Lp–H1 regularization, we will achieve some desired convergence rate by using the Bregman distance and some new source condition and nonlinearity condition.

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A Galerkin method for mixed parabolic–elliptic partial differential equations

Cited by 8 publications

References 21 publications

An inverse problem of identifying the diffusion coefficient in a coupled parabolic‐elliptic system

An inverse problem of identifying the diffusion coefficient in a coupled parabolic‐elliptic system

Uniform convergence analysis of hybrid numerical scheme for singularly perturbed problems of mixed type

Convergence rates of Tikhonov regularizations for parameter identification in a parabolic-elliptic system

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