Numerical Solution of the Poisson Equation on Domains with a Thin Layer of Random Thickness

Dambrine, Marc; Greff, Isabelle; Harbrecht, Helmut; Puig, Bénédicte

doi:10.1137/140998652

Cited by 17 publications

(28 citation statements)

References 24 publications

(43 reference statements)

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“…In the k−th step of Newton's method, knowing σ (k−1) , we solve (8) and (11) for u (k−1) , solve (40) for u , and then update σ (k) according to (37). In 1D, we have…”

Section: Resultsmentioning

confidence: 99%

“…When dealing with relatively small input variability and outputs that do not express high nonlinearity, perturbation type methods are most frequently used, where the random solutions are expanded via Taylor series around their mean and truncated at a certain order [21,11]. Typically, at most second-order expansion is used because the resulting system of equations are typically complicated beyond the second order.…”

Section: Introductionmentioning

confidence: 99%

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Estimation of exciton diffusion lengths of organic semiconductors in random domains

Chen

Zhang

et al. 2019

Journal of Computational Physics

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Exciton diffusion length plays a vital role in the function of opto-electronic devices. Oftentimes, the domain occupied by an organic semiconductor is subject to surface measurement error. In many experiments, photoluminescence over the domain is measured and used as the observation data to estimate this length parameter in an inverse manner based on the least square method. However, the result is sometimes found to be sensitive to the surface geometry of the domain. In this paper, we employ a random function representation for the uncertain surface of the domain. After non-dimensionalization, the forward model becomes a diffusion-type equation over the domain whose geometric boundary is subject to small random perturbations. We propose an asymptotic-based method as an approximate forward solver whose accuracy is justified both theoretically and numerically. It only requires solving several deterministic problems over a fixed domain. Therefore, for the same accuracy requirements we tested here, the running time of our approach is more than one order of magnitude smaller than that of directly solving the original stochastic boundary-value problem by the stochastic collocation method. In addition, from numerical results, we find that the correlation length of randomness is important to determine whether a 1D reduced model is a good surrogate for the 2D model. Subject class[2000] 34E05, 35C20, 35R60, 58J37, 65C99

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“…In the k−th step of Newton's method, knowing σ (k−1) , we solve (8) and (11) for u (k−1) , solve (40) for u , and then update σ (k) according to (37). In 1D, we have…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Estimation of exciton diffusion lengths of organic semiconductors in random domains

Chen

Zhang

et al. 2019

Journal of Computational Physics

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show abstract

“…[5] already derived the first three terms, u 0 , u 1 and u 2 . But the method we give below seems simpler and does not require the dilation technique and any asymptotic form for the differential operator L used in [5]. Actually, that kind of singular perturbation suits for the case that the solution itself develops a sharp peak in the thin layer, such as the traditional boundary layer analysis in fluid mechanics.…”

Section: The Elliptic Problem With Smooth Coefficientsmentioning

confidence: 99%

“…Note that although u [n] and v [n] have the same approximation order, there might still be a considerable difference in the accuracy of their approximation errors due to the effects of the prefactors. The numerical results in [5] show that the approximation u [n] produces much less accurate results than v [n] for n = 1, 2. This can be easily confirmed by the following simple onedimensional example:…”

Section: 2mentioning

confidence: 99%

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Asymptotic analysis for elliptic equations with small perturbations on domains in high-contrast medium

Chen

Zhang

et al. 2020

ASY

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We consider the asymptotic solutions of an interface problem corresponding to an elliptic partial differential equation with Dirichlet boundary condition and transmission condition, subject to the small geometric perturbation and the high contrast ratio of the conductivity. We consider two types of perturbations: the first corresponds to a thin layer coating a fixed bounded domain and the second is the perturbation of the interface. As the perturbation size tends to zero and the ratio of the conductivities in two subdomains tends to zero, the two-parameter asymptotic expansions on the fixed reference domain are derived to any order after the single parameter expansions are solved beforehand. Our main tool is the asymptotic analysis based on the Taylor expansions for the properly extended solutions on fixed domains. The Neumann boundary condition and Robin boundary condition arise in two-parameter expansions, depending on the relation of the geometric perturbation size and the contrast ratio.

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A domain mapping approach for elliptic equations posed on random bulk and surface domains

2020

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In this article, we analyse the domain mapping method approach to approximate statistical moments of solutions to linear elliptic partial differential equations posed over random geometries including smooth surfaces and bulk-surface systems. In particular, we present the necessary geometric analysis required by the domain mapping method to reformulate elliptic equations on random surfaces onto a fixed deterministic surface using a prescribed stochastic parametrisation of the random domain. An abstract analysis of a finite element discretisation coupled with a Monte-Carlo sampling is presented for the resulting elliptic equations with random coefficients posed over the fixed curved reference domain and optimal error estimates are derived. The results from the abstract framework are applied to a model elliptic problem on a random surface and a coupled elliptic bulk-surface system and the theoretical convergence rates are confirmed by numerical experiments.

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Numerical Solution of the Poisson Equation on Domains with a Thin Layer of Random Thickness

Cited by 17 publications

References 24 publications

Estimation of exciton diffusion lengths of organic semiconductors in random domains

Estimation of exciton diffusion lengths of organic semiconductors in random domains

Asymptotic analysis for elliptic equations with small perturbations on domains in high-contrast medium

A domain mapping approach for elliptic equations posed on random bulk and surface domains

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