Anderson Acceleration is an extrapolation technique used to accelerate the convergence of fixed-point iterations. It can post-process both Picard-like and Newton-like iterations, and when it works it can both accelerate and stabilize the solution process. We will look at some recent theoretical advances that describe how the method can be effective for finding solutions to discretized systems encountered in the approximation of nonlinear partial differential equations. Some numerical examples will demonstrate how recent advances in the theory are leading to more efficient and robust accelerated methods.

A numerical approximation is developed, analyzed, and investigated for quenching solutions to a degenerate Kawarada problem with a left and right Riemann-Liouville fractional Laplacian over a finite one dimensional domain. The numerical analysis provides criterion for the numerical approximations to be monotonic, nonnegative, and linearly stable throughout the computation. The numerical algorithm is used to develop an experimental scaling law relating the critical quenching domain size to the order of fractional derivative. Additional experiments indicate that imbalanced left and right derivative transport coefficients can attenuate or prevent quenching from occurring.

We consider flux-based multiple-porosity/multiple-permeability poroelasticity systems describing mulitple-network flow and deformation in a poro-elastic medium, also referred to as MPET models. The focus of the paper is on the convergence analysis of the fixed-stress split iteration, a commonly used coupling technique for the flow and mechanics equations defining poromechanical systems. We formulate the fixed-stress split method in this context and prove its linear convergence. The contraction rate of this fixed-point iteration does not depend on any of the physical parameters appearing in the model. This is confirmed by numerical results which further demonstrate the advantage of the fixed-stress split scheme over a fully implicit method relying on norm-equivalent preconditioning.

In this talk, we present a new finite element method based on flux variables. In many applications, the flux variables are often the quantity of interest. To approximate the flux variable accurately and efficiently, one transforms the second-order equations into a system of first-order and approximates both the primary and flux variables simultaneously. While this indeed produces accurate approximations for the flux variables, the resulting algebraic system is large and expensive to solve. We present a new method approximating the flux variables only without approximation of the primary variable. If necessary, the primary variable can be recovered from the flux approximation with the same order of accuracy. This new approach can be considered as a reduced version of the standard mixed finite element methods.