Data-sparse approximation to the operator-valued functions of elliptic operator

In previous papers the arithmetic of hierarchical matrices has been described, which allows us to compute the inverse, for instance, of finite element stiffness matrices discretising an elliptic operator $\mathcal{L}.$ The required computing time is up to logarithmic factors linear in the dimension of the matrix. In particular, this technique can be used for the computation of the discrete analogue of a resolvent $\left( zI-\mathcal{L}\right) ^{-1},$ $z\in\mathbb{C}.$

In the present paper, we consider various operator functions, the operator exponential $e^{-t\mathcal{L}},$ negative fractional powers ${\mathcal{L} }^{-\alpha}$, the cosine operator function $\cos(t\sqrt{\mathcal{L} })\mathcal{L}^{-k}$ and, finally, the solution operator of the Lyapunov equation. Using the Dunford-Cauchy representation, we get integrals which can be discretised by a quadrature formula which involves the resolvents $\left( z_{k}I-\mathcal{L}\right) ^{-1}$ mentioned above. We give error estimates which are partly exponentially, partly polynomially decreasing.