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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].

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We derive an approximate analytical solution, which describes the interface dynamics during the injection of supercritical carbon dioxide into homogeneous geologic media that are fully saturated with a host fluid. The host fluid can be either heavier (e.g., brine) or lighter (e.g., methane) than the injected carbon dioxide. Our solution relies on the Dupuit approximation and explicitly accounts for the buoyancy effects. The general approach is applicable to a variety of phenomena involving variable-density flows in porous media. In three dimensions under radial symmetry, the solution describes carbon dioxide injection; its two-dimensional counterpart can be used to model seawater intrusion into coastal aquifers. We conclude by comparing our solutions with existing analytical alternatives.

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