Applications of Anderson Acceleration to Algorithms for Newtonian and Non-Newtonian Fluid Simulation

After reviewing recent theoretical results for Anderson acceleration (AA), we consider its application to solving incompressible Navier-Stokes (NS) equations and regularized Bingham equations. For NS, the classical penalty method is considered, which typically will only work with very small penalty (but very small penalty causes issues with iterative solvers, making it not practical for large scale use). For regularized Bingham, we consider a Picard type iteration that has trouble converging for small regularization parameter. We show that both of these methods can be cast as fixed-point iterations that fall into the AA theory framework, which allows for improved convergence rates to be proven. Moreover, numerical results reveal that with AA, the classical penalty method is very effective even with O(1) penalty parameter and regularized Bingham Picard iteration is dramatically improved and nearly robust with respect to the regularization parameter.