Summary.
We develop a data-sparse and accurate approximation to parabolic solution operators in the case of a rather general elliptic part given by a strongly P-positive operator [4].
In the preceding papers [12]–[17], a class of matrices (\(\mathcal{H}\)-matrices) has been analysed which are data-sparse and allow an approximate matrix arithmetic with almost linear complexity. In particular, the matrix-vector/matrix-matrix product with such matrices as well as the computation of the inverse have linear-logarithmic cost. In the present paper, we apply the \(\mathcal{H}\)-matrix techniques to approximate the exponent of an elliptic operator.
Starting with the Dunford-Cauchy representation for the operator exponent, we then discretise the integral by the exponentially convergent quadrature rule involving a short sum of resolvents. The latter are approximated by the \(\mathcal{H}\)-matrices. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different time values. In the case of smooth data (coefficients, boundaries), we prove the linear-logarithmic complexity of the method.
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Received June 22, 2000 / Revised version received June 6, 2001 / Published online October 17, 2001
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Gavrilyuk, I., Hackbusch, W. & Khoromskij, B. $\mathcal{H}$-Matrix approximation for the operator exponential with applications. Numer. Math. 92, 83–111 (2002). https://doi.org/10.1007/s002110100360
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DOI: https://doi.org/10.1007/s002110100360