Wavelet compression of integral operators arising from boudary‐domain integral method

Abstract

AbstractThe boundary‐domain integral method uses Green's functions to write integral representations of partial differential equations. Since Green's functions are non‐local, the systems of linear equations arising from the discretization of integral representations are fully populated. Several algorithms have been proposed, which yield a data‐sparse approximation of these systems, such as the fast multipole method, adaptive cross approximation and among others, wavelet compression.In the framework of solving the Navier‐Stokes equations in velocity‐vorticity form one may utilize the boundary‐domain integral method for the solution of the kinematics equation to calculate the boundary vorticity values. Since the kinematics equation is a Poisson type equation, usually its integral representation is written with the Green's function for the Laplace operator. In this work, we introduce a false time into the equation and parabolize its nature. Thus, a time‐dependent Green's function may be used. This provides a new parameter, the time step, which can be set to control the Green's function. The time‐dependent Green's function is a global function, but by carefully choosing the time step, its behaviour is almost local. This makes it a good candidate for wavelet compression, yielding much better compression ratios at a given accuracy than when using the Green's function for the Laplace operator. However, as false time is introduced, several time steps must be solved in order to reach a converged solution. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)