Articlehybrid2020

A domain mapping approach for elliptic equations posed on random bulk and surface domains

Lewis Church · Ana Djurdjevac · Charles M. Elliott

13 citations0/0 files storedNumerische Mathematik

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Probabilistic and Robust Engineering Design Advanced Numerical Methods in Computational Mathematics Model Reduction and Neural Networks

Abstract

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.

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References

2010Spectral Methods for Uncertainty Quantification 2004Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations 2010Spectral methods for uncertainty quantification : with applications to computational fluid dynamics 2004Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations 2014An Introduction to Computational Stochastic PDEs 2011Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients 2014Stochastic finite element methods for partial differential equations with random input data 2006Numerical Methods for Differential Equations in Random Domains

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