Abstract
In this article, we provide a rigorous analysis of the solution to elliptic diffusion problems on random domains. In particular, based on the decay of the Karhunen-Loève expansion of the domain perturbation field, we establish decay rates for the derivatives of the random solution that are independent of the stochastic dimension. For the implementation of a related approximation scheme, like quasi-Monte Carlo quadrature, stochastic collocation, etc., we propose parametric finite elements to compute the solution of the diffusion problem on each individual realization of the domain generated by the perturbation field. This simplifies the implementation and yields a non-intrusive approach. Having this machinery at hand, we can easily transfer it to stochastic interface problems. The theoretical findings are complemented by numerical examples for both, stochastic interface problems and boundary value problems on random domains.
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Notes
It is sufficient to assume that \(\mathbf{V}\) is a \(C^1\)-diffeomorphism and satisfies the uniformity in \({C^1(\overline{D_{\mathrm{ref}}};\mathbb {R}^d)}\). Nevertheless, in order to obtain \(H^2\)-regularity of the model problem, we make this stronger assumption.
With “formally” we mean that we ignore here the fact that the product of matrices is in general not Abelian. Nevertheless, a differentiation yields exactly the appearing products in a permuted order. The formal representation is justified since we only consider the norm of the representation in the sequel.
Each node consists of two quad-core Intel(R) Xeon(R) X5550 CPUs with a clock rate of 2.67GHz (hyperthreading enabled) and 48GB of main memory.
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This research has been supported by the Swiss National Science Foundation (SNSF) through the project “Rapid Solution of Boundary Value Problems on Stochastic Domains”.
Appendix
Appendix
Lemma 8
Let \({\varvec{\gamma }}=\{\gamma _k\}_k\in \ell ^1(\mathbb {N})\) with finite support \(\mathcal {I}\subset \mathbb {N}\) and \(\gamma _k\ge 0\). Moreover, assume that \(c_{\varvec{\gamma }}\mathrel {\mathrel {\mathop :}=}\sum _{k\in \mathcal {I}}\gamma _k<1\). Then, it holds
and therefore there exists a constant with \(|{\varvec{\alpha }}|!/{\varvec{\alpha }}!{\varvec{\gamma }}^{\varvec{\alpha }}\le c\) for all \({\varvec{\alpha }}\in \mathbb {N}^M_0\), where we set \(M\mathrel {\mathrel {\mathop :}=}|\mathcal {I}|\) and \(0^0=1\).
Proof
It holds
by the multinomial theorem and the limit of the geometric series. \(\Box \)
Lemma 9
Let \(c,m\in \mathbb {R}\) with \(m\ge 2\) and \(c\ge m/(m-1)\). It holds for \(n\in \mathbb {N}\) that
Proof
It holds
Omitting the second summand together with the condition \(c\ge m/(m-1)\) yields the assertion. \(\Box \)
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Harbrecht, H., Peters, M. & Siebenmorgen, M. Analysis of the domain mapping method for elliptic diffusion problems on random domains. Numer. Math. 134, 823–856 (2016). https://doi.org/10.1007/s00211-016-0791-4
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DOI: https://doi.org/10.1007/s00211-016-0791-4