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Elliptic boundary value problems which are posed on a random domain can bemapped to a fixed, nominal domain. The randomness is thus transferred to thediffusion matrix and the loading. While this domain mapping method is quiteefficient for theory and practice, since only a single domain discretisation isneeded, it also requires the knowledge of the domain mapping.  However, in certain applications, the random domain is only described by itsrandom boundary, while the quantity of interest is defined on a fixed,deterministic subdomain. In this setting, it thus becomes necessary to computea random domain mapping on the whole domain, such that the domain mapping isthe identity on the fixed subdomain and maps the boundary of the chosen fixed,nominal domain on to the random boundary.  To overcome the necessity of computing such a mapping, we therefore couplethe finite element method on the fixed subdomain with the boundary elementmethod on the random boundary. We verify the required regularity of thesolution with respect to the random domain mapping for the use of multilevelquadrature, derive the coupling formulation, and show by numerical results thatthe approach is feasible.

Multilevel quadrature for elliptic problems on random domains by the coupling of FEM and BEM