Ghosh, Aditi (2013-10). Fast Algorithms for Biharmonic Problems and Applications to Fluid Dynamics. Doctoral Dissertation.
Many areas of physics, engineering and applied mathematics require solutions of inhomogeneous biharmonic problems. For example, various problems on Stokes flow and elasticity can be cast into biharmonic boundary value problems. Hence the slow viscous flow problems are generally modeled using biharmonic boundary value problems which have widespread applications in many areas of industrial problems such as flow of molten metals, flow of particulate suspensions in bio-fluid dynamics, just to mention a few. In this dissertation, we derive, implement, validate, and apply fast and high order accurate algorithms to solve Poisson problems and inhomogeneous biharmonic problems in the interior of a unit disc in the complex plane. In particular, we use two methods to solve inhomogeneous biharmonic problems: (i) the double-Poisson method which is based on transforming biharmonic problems into solving a sequence of Poisson problems (sometime also one homogeneous biharmonic problem) and then making use of the fast Poisson solver developed in this dissertation.; (ii) the direct method which uses the fast biharmoninc solver also developed in this dissertation. Both of these methods are analyzed for accuracy, complexity and efficiency. These biharmonic solvers have been compared with each other and have been applied to solve several Stokes flow problems and elasticity problems. The fast Poisson algorithm is derived here from exact analysis of the Green's function formulation in the complex plane. This algorithm is essentially a recast of the fast Poisson algorithm of Borges and Daripa from the real plane to the complex plane. The fast biharmonic algorithms for several boundary conditions for use in the direct method mentioned above have been derived in this dissertation from exact analysis of the representation of their solutions in terms of problem specific Green's function in the complex plane. The resulting algorithms primarily use fast Fourier transforms and recursive relations in Fourier space. The algorithms have been analyzed for their accuracy, complexity, efficiency, and subsequently tested for validity against several benchmark test problems. These algorithms have an asymptotic complexity of O(log N ) per degree of freedom with very low constant which is hidden behind the order estimate. The direct and double-Poisson methods have been applied to solving the steady, incompressible slow viscous flow problem in a cir- cular cylinder and some problems from elasticity. The numerical results from these computations agree well with existing results on these problems.