Numerical Methods for Differential Equations in Random Domains

Authors:
Dongbin Xiu;Daniel M. Tartakovsky
Affiliations:
-;-
Venue:
SIAM Journal on Scientific Computing
Year:
2006

Citing 0
Cited 9

A finite element method for elliptic problems with stochastic input data

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Uncertainty quantification via random domain decomposition and probabilistic collocation on sparse grids

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Iterative Solvers for the Stochastic Finite Element Method

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Original article: On an approach to deal with Neumann boundary value problems defined on uncertain domains: Numerical experiments

Mathematics and Computers in Simulation
A new level-set based approach to shape and topology optimization under geometric uncertainty

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Adaptive ANOVA decomposition of stochastic incompressible and compressible flows

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Wave scattering by randomly shaped objects

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Error analysis of generalized polynomial chaos for nonlinear random ordinary differential equations

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Extended stochastic FEM for diffusion problems with uncertain material interfaces

Computational Mechanics

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Abstract

Physical phenomena in domains with rough boundaries play an important role in a variety of applications. Often the topology of such boundaries cannot be accurately described in all of its relevant detail due to either insufficient data or measurement errors or both. This topological uncertainty can be efficiently handled by treating rough boundaries as random fields, so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains. To deal with this class of problems, we propose a novel computational framework, which is based on using stochastic mappings to transform the original deterministic/stochastic problem in a random domain into a stochastic problem in a deterministic domain. The latter problem has been studied more extensively, and existing analytical/numerical techniques can be readily applied. In this paper, we employ both a stochastic Galerkin method and Monte Carlo simulations to solve the transformed stochastic problem. We demonstrate our approach by applying it to an elliptic problem in single- and double-connected random domains, and comment on the accuracy and convergence of the numerical methods.