Anderson acceleration for a regularized Bingham model

TL;DR

This paper introduces an Anderson acceleration-based nonlinear solver for the regularized Bingham equations, demonstrating improved convergence and robustness in finite element simulations of viscoplastic flow in 2D and 3D.

Contribution

The paper develops and analyzes a novel Anderson acceleration method applied to a Picard iteration for the regularized Bingham model, enhancing solver efficiency and robustness.

Findings

Accelerated convergence of the nonlinear solver using Anderson acceleration.

Solver remains effective as the regularization parameter approaches zero.

Numerical results confirm spatial convergence and robustness in 2D and 3D simulations.

Abstract

This paper studies a finite element discretization of the regularized Bingham equations that describe viscoplastic flow. An efficient nonlinear solver for the discrete model is then proposed and analyzed. The solver is based on Anderson acceleration (AA) applied to a Picard iteration, and we show accelerated convergence of the method by applying AA theory (recently developed by the authors) to the iteration, after showing sufficient smoothness properties of the associated fixed point operator. Numerical tests of spatial convergence are provided, as are results of the model for 2D and 3D driven cavity simulations. For each numerical test, the proposed nonlinear solver is also tested and shown to be very effective and robust with respect to the regularization parameter as it goes to zero.