A novel deflation method to solve the 3-D discontinuous
and singular Poisson equation

Jok M. Tang
Dept. of Applied Mathematical Analysis
Delft University of Technology
Mekelweg 4, 2628 CD Delft, The Netherlands
j.m.tang@ewi.tudelft.nl

Recently, simulating bubbly flows is a very popular topic in CFD. These bubbly flows are governed by the Navier-Stokes equations. In many popular operator splitting formulations for these equations, solving the linear system coming from the discontinuous Poisson equation takes the most computational time, despite of its elliptic origins. ICCG is widely used for this purpose, but for complex bubbly flows this method shows slow convergence.

Moreover, new insights are given into the properties of invertible and singular deflated and preconditioned linear systems, where the coefficient matrices are symmetric and positive (semi-) definite. These linear systems can be derived from a discretization of the Poisson equation with Neumann boundary conditions. Sometimes these linear systems are forced to be invertible leading to a worse (effective) condition number. If ICCG is used to solve this problem, the convergence is significantly slower than for the case of the original singular problem.

We show that applying the deflation technique, which leads to the DICCG method, remedies the worse condition number and the worse convergence of ICCG. Moreover, some useful equalities are derived from the deflated variants of the singular and invertible matrices, which are also generalized to preconditioned methods. It appears that solving the invertible and singular linear systems with DICCG leads to exactly the same convergence results.

This new method DICCG incorporates the eigenmodes corresponding to the components which caused the slow convergence of ICCG. Coarse linear systems have to be solved within DICCG. We discuss some methods to do this efficiently which results in two approaches DICCG1 and DICCG2.

Thereafter we show with numerical experiments that both DICCG approaches are very efficient and they emphasize also the theoretical results. Compared to ICCG, DICCG decreases significantly the number of iterations and the computational time as well which are required for solving Poisson equation in applications of 2-D and 3-D bubbly flows.