H - Matrix Approximation for Elliptic Solution Operators inCylindric

We develop a data-sparse and accurate approximation of the normalised hyperbolic operator sine family generated by a strongly P-positive elliptic operator deened in 4, 7]. In the preceding papers 14]-18], a class of H-matrices has been analysed which are data-sparse and allow an approximate matrix arithmetic with almost linear complexity. An H-matrix approximation to the operator exponent with a strongly P-positive operator was proposed in 5]. In the present paper, we apply the H-matrix techniques to approximate the elliptic solution operator on cylindric domains a; b] associated with the elliptic operator d 2 dx 2 ? L, x 2 a; b]. It is explicitly presented by the operator-valued normalised hyperbolic sine function sinh ?1 (p L)sinh(x p L) of an elliptic operator L deened in. Starting with the Dunford-Cauchy representation for the hyperbolic sine operator, we then discretise the integral by the exponentially convergent quadrature rule involving a short sum of resolvents. The latter are approximated by the H-matrix techniques. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of diierent values of the spatial variable x 2 a; b]. The approach is applied to elliptic partial diierential equations in domains composed of rectangles or cylinders. In particular, we consider the H-matrix approximation to the interface Poincar e-Steklov operators with application in the Schur-complement domain decomposition method.