Direct numerical identification of boundary values in the Laplace equation

Hayashi, Keisuke; Onishi, Kazuei; Ohura, Yoko

doi:10.1016/s0377-0427(02)00703-3

Cited by 6 publications

(4 citation statements)

References 8 publications

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“…Note that our study remains valid for any domain with a smooth enough boundary 2 which can be written, via a regular transformation, in form (17). In particular, it remains valid for the cases where 2 is a side of a rectangle or an arc of a circle, which are met most frequently for the numerical test examples taken in the literature [5][6][7][8][9][10]18].…”

Section: Approximation Of the Problemmentioning

confidence: 88%

“…In the past few decades, several iterative methods aiming at the approximation of the solution of different variants of this problem were presented. Some authors were interested in the mathematical analysis of their algorithms [3,5,7,8,10]; others studied the numerical aspect (implementation, stability, etc) [6,9,13,[18][19][20]. But often, the convergence of a sequence of solutions of the discrete problems to the solution of the continuous one remains without theoretical justification.…”

Section: Introductionmentioning

confidence: 99%

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Convergence analysis for finite element approximation to an inverse Cauchy problem

Chakib

Nachaoui

2006

Inverse Problems

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In this paper, we propose an approximate optimal control formulation of the Cauchy problem, for an elliptic equation, equivalent to the original one under some regularity assumptions on the data. We prove the existence of a solution and show the convergence of a subsequence of approximate solutions, obtained via finite element approximation, to a solution of the continuous problem. Finally, we give some numerical results showing the efficiency of our approach and confirming the convergence result.

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Section: Approximation Of the Problemmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Convergence analysis for finite element approximation to an inverse Cauchy problem

Chakib

Nachaoui

2006

Inverse Problems

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“…Dosiyev and Buranay [28] presented a one-block method to compute generalized stress intensity factors for Laplace's equation on a square with a slit and on an L-shaped domains. Direct numerical identification of boundary values in the Laplace equation is investigated by Hayashi et al [29]. Jorge et al [30] focused on self-regular boundary integral equation formulations for Laplace's equation in a 2D domain.…”

Section: Introductionmentioning

confidence: 99%

Numerical calculations of the potential on the rectangular and ellpitic domains with various aspect ratios

Panahi¹,

Ghassemi²,

Nowruzi³

2017

J. Appl. Math. Comput. Mech.

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Abstract. In the current study, the Laplace equation is solved for rectangular and elliptical computational domains by using the boundary element method (BEM). For this accomplishment, 120 different aspect ratios in a rectangular and elliptical computational domains are designed. The Dirichlet and Neumann boundary conditions are used respectively for a rectangular and elliptical domains. Also, the Gaussian quadrature integral method is applied to solve the influence coefficient matrix in BEM. To assess a different aspect ratio on the potential solution, two different measurement positions are intended. According to our finding, with an increase of the aspect ratio, the potential value is increased for both rectangular and elliptical domains. However, a potential increment with aspect ratio enhancement is more visible in the elliptical domain.MSC 2010:76M15, 76B07

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“…The influence of measurement location, measurement error and element option were investigated. An inverse problem for Laplace equations was recast into primary and adjoint boundary value problems by Hayashi et al [8,9]. The Dirichlet and Neumann data were specified on respective part of the boundary, while no data on the second part of the boundary were given and Robin condition was prescribed on the third part of the boundary.…”

Section: Introductionmentioning

confidence: 99%

The PCGM for Cauchy inverse problems in 3D potential

Zhou¹,

Bian²,

Cheng³

et al. 2013

Boundary Elements and Other Mesh Reduction Methods XXXVI

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The preconditioned conjugate gradient method (PCGM) in combination with the boundary element method (BEM) is developed for reconstructing the boundary conditions in 3-D potential. Morozov's discrepancy principle is employed to select the iteration step. The semi-analytical integral algorithm is proposed to treat the nearly singular integrals when the interior points are very close to the boundary. The numerical results confirm that the PCGM produces convergent and stable numerical solutions with respect to decreasing the amount of noise added into the input data. The numerical solutions are sensitive to the locations of the interior points when these points are distributed near the boundary without boundary conditions. The results are more accurate when these points are closer to the boundary.

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Direct numerical identification of boundary values in the Laplace equation

Cited by 6 publications

References 8 publications

Convergence analysis for finite element approximation to an inverse Cauchy problem

Convergence analysis for finite element approximation to an inverse Cauchy problem

Numerical calculations of the potential on the rectangular and ellpitic domains with various aspect ratios

The PCGM for Cauchy inverse problems in 3D potential

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