The FFTRR-based fast decomposition methods for solving complex biharmonic problems and incompressible flows

Abstract

In this work we present several computational results on the complex biharmonic problems. First, we derive FFTRR based fast algorithms for solving Dirichlet and Neumann type complex Poisson problems in the complex plane. These are based on the use of fast Fourier transform (FFT), analysis based recursive relations (RR) in Fourier space, and high order quadrature methods. Our second result is the application of these fast Poisson algorithms to solving four types of inhomogeneous biharmonic problems in the complex plane using decomposition methods. Lastly, we apply these high order accurate fast algorithms for the complex inhomogeneous biharmonic problems to solving Stokes flow problems at low and moderate Reynolds number. All these algorithms are inherently parallelizable though only sequential implementations have been performed. These algorithms have theoretical complexity of the order O(logN) per grid point where N is the total number of grid points in the discretization of the domain. These algorithms have many other desirable features some of which are discussed in the paper. Numerical results have been presented which show performance of these algorithms.