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

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.