The FFTRR-based fast direct algorithms for complex inhomogeneous biharmonic problems with applications to incompressible flows
Academic Article
Overview
Research
Identity
Additional Document Info
Other
View All
Overview
abstract
2016, Springer Science+Business Media New York. We develop analysis-based fast and accurate direct algorithms for several biharmonic problems in a unit disk derived directly from the Greens functions of these problems and compare the numerical results with the decomposition algorithms (see Ghosh and Daripa, IMA J. Numer. Anal. 36(2), 824850 [17]) in which the biharmonic problems are first decomposed into lower order problems, most often either into two Poisson problems or into two Poisson problems and a homogeneous biharmonic problem. One of the steps in the decomposition algorithm as discussed in Ghosh and Daripa (IMA J. Numer. Anal. 36(2), 824850 [17]) for solving certain biharmonic problems uses the direct algorithm without which the problem can not be solved. Using classical Greens function approach for these biharmonic problems, solutions of these problems are represented in terms of singular integrals in the complex zplane (the physical plane) involving explicitly the boundary conditions. Analysis of these singular integrals using FFT and recursive relations (RR) in Fourier space leads to the development of these fast algorithms which are called FFTRR based algorithms. These algorithms do not need to do anything special to overcome coordinate singularity at the origin as often the case when solving these problems using finite difference methods in polar coordinates. These algorithms have some other desirable properties such as the ease of implementation and parallel in nature by construction. Moreover, these algorithms have O(logN) complexity per grid point where N2 is the total number of grid points and have very low constant behind this order estimate of the complexity. Performance of these algorithms is shown on several test problems. These algorithms are applied to solving viscous flow problems at low and moderate Reynolds numbers and numerical results are presented.
