An ultraweak DPG method for viscoelastic fluids

TL;DR

This paper applies an ultraweak DPG finite element method to viscoelastic fluid flow problems, demonstrating its stability, adaptive meshing capabilities, and computational efficiency for complex models.

Contribution

It develops an intrinsic a posteriori error indicator for adaptive mesh refinement in DPG methods applied to viscoelastic fluids, highlighting the method's stability and efficiency.

Findings

The DPG method is inherently stable without additional stabilization.

The method enables parameter-specific adaptive mesh generation.

The symmetric positive definite stiffness matrix allows efficient direct solvers.

Abstract

We explore a vexing benchmark problem for viscoelastic fluid flows with the discontinuous Petrov-Galerkin (DPG) finite element method of Demkowicz and Gopalakrishnan [1,2]. In our analysis, we develop an intrinsic a posteriori error indicator which we use for adaptive mesh generation. The DPG method is useful for the problem we consider because the method is inherently stable---requiring no stabilization of the linearized discretization in order to handle the advective terms in the model. Because stabilization is a pressing issue in these models, this happens to become a very useful property of the method which simplifies our analysis. This built-in stability at all length scales and the a posteriori error indicator additionally allows for the generation of parameter-specific meshes starting from a common coarse initial mesh. A DPG discretization always produces a symmetric positive definite stiffness matrix. This feature allows us to use the most efficient direct solvers for all of our computations. We use the Camellia finite element software package [3,4] for all of our analysis.