Numerical Methods for Differential Equations in Random Domains

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.1. Introduction. Physical phenomena described by differential equations in domains with rough geometries are ubiquitous and include applications ranging from surface imaging [24] to manufacturing of nano-devices [5]. Indeed, given a proper spatial resolution, virtually any natural or manufactured surface becomes rough. Consequently, there is a growing interest in experimental, theoretical, and numerical studies of deterministic and probabilistic descriptions of such surfaces and of solutions of differential equations defined on the resulting domains.The early attempts to represent surface roughness and to study its effects on system behavior were based on simplified, easily parameterizable, deterministic surface inhomogeneities, such as symmetrical asperities to represent indentations [32,26] and semispheres to represent protrusions [17]. More general representations of surface roughness, which nevertheless admit simple parameterizations, include sinusoidal corrugations [13] and periodic linear segments [6]. A recent trend in describing surface roughness is to use fractal and fractal-like representation. These include deterministic Von Koch's [10, 6, 8] and Minkowski's [6] fractals, as well as their random generalizations [29].

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

“…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…”

Approximation Methods to Solve Stochastic Problems in Computational Electromagnetics

“…The most natural way to account for randomness on the geometry consist in remeshing according to the deformation but the remeshing leads to a discontinuous solution in the space of the input parameters and can create additional numerical noise which can disturb the random solution. Alternatives have been proposed in the literature [5][6][7][8][9] to avoid remeshing. In the following, we will focus mainly on uncertainties on the behaviour laws.…”

“…A stochastic partial differential equation system is generally numerically solved by applying, like in the deterministic case (see 2.1), a semi-discretisation in space (see 6). The DoF's a i of the vector potential (see (5)), which were real numbers in the deterministic case, becomes random variables a i [p(θ )]. The matrix S and the vector F have random entries s i j [p(θ )] and f i [p(θ )] and the unknown vector A is random and satisfies:…”

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

“…We study the effects of this uncertainty and impose the boundary condition at stochastically varying positions in space. 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].…”