An a posteriori error estimator based on least-squares finite element solutions for viscoelastic fluid flows

Hsueh-Chen Lee; Hyesuk Lee; Hsueh-Chen Lee; Hyesuk Lee

doi:10.3934/era.2021012

Electronic Research Archive

2021, Volume 29, Issue 4: 2755-2770. doi: 10.3934/era.2021012

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An a posteriori error estimator based on least-squares finite element solutions for viscoelastic fluid flows

Hsueh-Chen Lee ^{1
,
,},
Hyesuk Lee ^{2
,}

1.
Center for General Education, Wenzao Ursuline University of Languages, 900 Mintsu 1st Road Kaohsiung, Taiwan
2.
School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC 29634-0975, USA

Received: 01 July 2020 Revised: 01 December 2020 Published: 22 February 2021
Primary: 65N30, 65N50, Secondary: 76A10

We present a posteriori error estimator strategies for the least-squares finite element method (LS) to approximate the exponential Phan-Thien-Tanner (PTT) viscoelastic fluid flows. The error estimator provides adaptive mass weights and mesh refinement criteria for improving LS solutions using lower-order basis functions and a small number of elements. We analyze an a priori error estimate for the first-order linearized LS system and show that the estimate is supported by numerical results. The LS approach is numerically tested for a convergence study and then applied to the flow past a slot channel. Numerical results verify that the proposed approach improves numerical solutions and resolves computational difficulties related to the presence of corner singularities and limitations arising from the exorbitant number of unknowns.
- Least-squares,
- viscoelastic fluid,
- adaptive mesh refinement,
- a posteriori error estimator,
- transverse slot
Citation: Hsueh-Chen Lee, Hyesuk Lee. An a posteriori error estimator based on least-squares finite element solutions for viscoelastic fluid flows[J]. Electronic Research Archive, 2021, 29(4): 2755-2770. doi: 10.3934/era.2021012

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Abstract

We present a posteriori error estimator strategies for the least-squares finite element method (LS) to approximate the exponential Phan-Thien-Tanner (PTT) viscoelastic fluid flows. The error estimator provides adaptive mass weights and mesh refinement criteria for improving LS solutions using lower-order basis functions and a small number of elements. We analyze an a priori error estimate for the first-order linearized LS system and show that the estimate is supported by numerical results. The LS approach is numerically tested for a convergence study and then applied to the flow past a slot channel. Numerical results verify that the proposed approach improves numerical solutions and resolves computational difficulties related to the presence of corner singularities and limitations arising from the exorbitant number of unknowns.

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