A domain mapping approach for elliptic equations posed on random bulk and surface domains
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Numerische Mathematik
A domain mapping approach for elliptic equations posed on random bulk and surface domains Lewis Church1 · Ana Djurdjevac2 · Charles M. Elliott1 Received: 2 May 2019 / Revised: 9 April 2020 © The Author(s) 2020
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. Keywords Surface finite element method · Domain mapping method · Random domains · Random elliptic equations
The work of L. Church was supported by EPSRC as part of the MASDOC DTC, Grant No. EP /HO23364/1. The work of A. Djurdjevac was supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via MATH+: The Berlin Mathematics Research Center, EXC-2046/1 a Project ID: 390685689. The work of C. M. Elliott was supported by a Royal Society Wolfson Research Merit Award.
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Lewis Church [email protected] Ana Djurdjevac [email protected] Charles M. Elliott [email protected]
1
Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK
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Institute of Mathematics, Strae des 17. Juni 136, Technical University of Berlin, 10623 Berlin, Germany
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L. Church et al.
Mathematics Subject Classification 65N12 · 65N30 · 65C05
1 Introduction In the mathematical characterization of numerous scientific and engineering systems, the topology of the domain may not be precisely described. The main sources of uncertainty are usually insufficient data, measurement errors or manufacturing variability. This uncertainty in the geometry often naturally appears in many applications including surface imaging, manufacturing of nano-devices, material science and biological systems. As a result, the analysis of uncertainty in the computational domain has become an interesting and rich mathematical field. 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 o
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