Abstract
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.
Highlights
In the mathematical characterization of numerous scientific and engineering systems, the topology of the domain may not be precisely described
A comprehensive summary concerning the first directions in the treatment of elliptic partial differential equations (PDEs) in random domains can be found in [4,8,19,27,31] and recently [11] for a parabolic equation on a randomly evolving domain
Aside from the fictitious domain method [4,26,27], the main approaches utilize a probabilistic framework by describing the random boundary of the domain with a random field
Summary
In the mathematical characterization of numerous scientific and engineering systems, the topology of the domain may not be precisely described. The key idea behind this method is to define an extension of the random boundary process into the interior domain to form a complete random mapping for the whole domain and to use this domain mapping to transform the original partial differential equation on the random domain onto the fixed deterministic reference domain resulting in partial differential equations with random coefficients. For the latter formulation, there is a wealth of literature available on numerical techniques to compute any quantities of interest, see for example [17, 23,24].
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