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

IMA JOURNAL OF numerical ANALYSIS 2016年第2期36卷 824-850页

作者： Ghosh, Aditi Daripa, Prabir Texas A&M Univ Dept Math College Stn TX 77843 USA

In this work, we present several computational results on the complex biharmonic problems. First, we derive fast Fourier transform recursive relation (FFTRR)-based fast algorithms for solving Dirichlet-and Neumann-type complex Poisson problems in the complex plane. These are based on the use of FFT, analysis-based RRs 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 (log N) per grid point, where N-2 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.

关键词： complex Biharmonic equation complex Poisson equation fast algorithms stokes equations incompressible flows FFT recursive relations numerical implementation in matlab

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The FFTRR-based fast direct algorithms for complex inhomogeneous biharmonic problems with applications to incompressible flows

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numerical ALGORITHMS 2017年第4期75卷 937-971页

作者： Daripa, Prabir Ghosh, Aditi Texas A&M Univ Dept Math College Stn TX 77843 USA

We develop analysis-based fast and accurate direct algorithms for several biharmonic problems in a unit disk derived directly from the Green's functions of these problems and compare the numerical results with the "decomposition" algorithms (see Ghosh and Daripa, IMA J. Numer. Anal. 36(2), 824-850 [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), 824-850 [17]) for solving certain biharmonic problems uses the "direct algorithm" without which the problem can not be solved. Using classical Green's function approach for these biharmonic problems, solutions of these problems are represented in terms of singular integrals in the complex z-plane (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 N (2) 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.

关键词： Complex Biharmonic equation Green's function Fast algorithms Stokes equations Steady incompressible flows FFT Recursive relations numerical implementation in matlab

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