Gaussian spectral rules for second order finite-difference schemes

Vladimir Druskin

Schlumberger-Doll Research, Old Quarry Rd., Ridgefield CT 06877-4108


The subject of this talk is targeted grid optimization for second order FD approximations of elliptic and hyperbolic problems arising in remote sensing, control theory, etc., where the solution is needed only at few receiver points. The optimization can be viewed as an extension of the conception of the Gaussian quadrature rules to the second order finite-difference schemes. A standard Gaussian k-point quadrature for numerical integration is chosen to be exact for 2k polynomials, and an optimal grid with k nodes is chosen to match the impedance at the receiver points for some 2k frequencies. To solve this problem we employ methods of rational approximation and linear algebra traditionally used for optimization of iterative methods. The optimization yields exponential convergence of the impedance, i.e., the standard second order scheme with the three-point stencil exhibits spectral superconvergence. The optimized scheme is applied to two- and three- dimensional problems in electromagnetic and acoustic well logging. Our numerical experiments exhibit exponential superconvergence at prescribed points (receivers), where the cost per grid node is close to that of the standard second order finite-difference scheme. We observe more than one order speedup for practically important problems. Collaborators: Sergey Asvadurov (SLB), David Ingerman (Princeton-MIT), Shari Moskow (UFL) and Leonid Knizhnerman (CGE).