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arXiv · 1511.02626

$\mathcal{H}$-matrix based second moment analysis for rough random fields and finite element discretizations

Jürgen Dölz·Helmut Harbrecht·Michael D. Peters
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Abstract

We consider the efficient solution of strongly elliptic partial differential equations with random load based on the finite element method. The solution's two-point correlation can efficiently be approximated by means of an $\mathcal{H}$-matrix, in particular if the correlation length is rather short or the correlation kernel is non-smooth. Since the inverses of the finite element matrices which correspond to the differential operator under consideration can likewise efficiently be approximated in the $\mathcal{H}$-matrix format, we can solve the correspondent $\mathcal{H}$-matrix equation in essentially linear time by using the $\mathcal{H}$-matrix arithmetic. Numerical experiments for three-dimensional finite element discretizations for several correlation lengths and different smoothness are provided. They validate the presented method and demonstrate that the computation times do not increase for non-smooth or shortly correlated data.

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Jürgen Dölz, Helmut Harbrecht, Michael D. Peters. 2015-11-09. $\mathcal{H}$-matrix based second moment analysis for rough random fields and finite element discretizations. https://arxiv.org/abs/1511.02626

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