A fictitious domain approach to the numerical solution of PDEs in stochastic domains

Canuto, Claudio; Kozubek, Tomáš

doi:10.1007/s00211-007-0086-x

Cited by 86 publications

(86 citation statements)

References 32 publications

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“…Especially, the treatment of uncertainties in the computational domain has become of growing interest, see e.g. [5,18,33,36]. In this article, we consider the elliptic diffusion equation…”

Section: Introductionmentioning

confidence: 99%

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Analysis of the domain mapping method for elliptic diffusion problems on random domains

2016

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In this article, we provide a rigorous analysis of the solution to elliptic diffusion problems on random domains. In particular, based on the decay of the Karhunen-Loève expansion of the domain perturbation field, we establish decay rates for the derivatives of the random solution that are independent of the stochastic dimension. For the implementation of a related approximation scheme, like quasi-Monte Carlo quadrature, stochastic collocation, etc., we propose parametric finite elements to compute the solution of the diffusion problem on each individual realization of the domain generated by the perturbation field. This simplifies the implementation and yields a non-intrusive approach. Having this machinery at hand, we can easily transfer it to stochastic interface problems. The theoretical findings are complemented by numerical examples for both, stochastic interface problems and boundary value problems on random domains.

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“…Especially, the treatment of uncertainties in the computational domain has become of growing interest, see e.g. [5,18,33,36]. In this article, we consider the elliptic diffusion equation…”

Section: Introductionmentioning

confidence: 99%

“…tomography. Besides the fictitious domain approach considered in [5], one might essentially distinguish two approaches: the perturbation method and the domain mapping method.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the domain mapping method for elliptic diffusion problems on random domains

2016

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“…Related techniques are boundary perturbation [26], Lagrangian approach [1] and isoparametric mapping [5]. Other techniques dealing with geometric uncertainty include polynomial chaos with remeshing of geometry [8,9] as well as chaos collocation methods with fictious domains [3,17].…”

Section: Introductionmentioning

confidence: 99%

The effect of uncertain geometries on advection–diffusion of scalar quantities

Wahlsten

Nordström

2017

Bit Numer Math

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The two dimensional advection-diffusion equation in a stochastically varying geometry is considered. The varying domain is transformed into a fixed one and the numerical solution is computed using a high-order finite difference formulation on summation-by-parts form with weakly imposed boundary conditions. Statistics of the solution are computed non-intrusively using quadrature rules given by the probability density function of the random variable. As a quality control, we prove that the continuous problem is strongly well-posed, that the semi-discrete problem is strongly stable and verify the accuracy of the scheme. The technique is applied to a heat transfer problem in incompressible flow. Statistical properties such as confidence intervals and variance of the solution in terms of two functionals are computed and discussed. We show that there is a decreasing sensitivity to geometric uncertainty as we gradually lower the frequency and amplitude of the randomness. The results are less sensitive to variations in the correlation length of the geometry.

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“…Especially, the treatment of uncertainties in the computational domain has become of growing interest, see e.g. [5,23,38,41]. In this article, we consider the elliptic diffusion equation as a model problem where the underlying domain D(ω) ⊂ R d or respectively its boundary ∂ D(ω) are random.…”

Section: Introductionmentioning

confidence: 99%

Combination Technique Based Second Moment Analysis for Elliptic PDEs on Random Domains

Harbrecht

Peters

2016

Lecture Notes in Computational Science and Engineering

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In this article, we propose the sparse grid combination technique for the second moment analysis of elliptic partial differential equations on random domains. By employing shape sensitivity analysis, we linearize the influence of the random domain perturbation on the solution. We derive deterministic partial differential equations to approximate the random solution's mean and its covariance with leading order in the amplitude of the random domain perturbation. The partial differential equation for the covariance is a tensor product Dirichlet problem which can efficiently be determined by Galerkin's method in the sparse tensor product space. We show that this Galerkin approximation coincides with the solution derived from the combination technique provided that the detail spaces in the related multiscale hierarchy are constructed with respect to Galerkin projections. This means that the combination technique does not impose an additional error in our construction. Numerical experiments quantify and qualify the proposed method.

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A fictitious domain approach to the numerical solution of PDEs in stochastic domains

Cited by 86 publications

References 32 publications

Analysis of the domain mapping method for elliptic diffusion problems on random domains

Analysis of the domain mapping method for elliptic diffusion problems on random domains

The effect of uncertain geometries on advection–diffusion of scalar quantities

Combination Technique Based Second Moment Analysis for Elliptic PDEs on Random Domains

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