An ultraweak DPG method for viscoelastic fluids

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.

arXiv comment20 pages, 18 figures, 6 tables

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• Fully-implicit log-conformation formulation of constitutive laws
• The DPG methodology applied to different variational formulations of linear elasticity
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Cited by in corpus (5)

• High-order polygonal discontinuous Petrov-Galerkin (PolyDPG) methods using ultraweak formulations
• Discrete least-squares finite element methods
• Goal-oriented adaptive mesh refinement for non-symmetric functional settings
• Using a DPG method to validate DMA experimental calibration of viscoelastic materials
• Numerical Optimization of Planar Nozzle Shapes for Fused Deposition Modeling