# American Institute of Mathematical Sciences

doi: 10.3934/era.2021012

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

 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

* Corresponding author: Hsueh-Chen Lee

The first author of this article would like to dedicate this article to her advisor Professor Tsu-Fen Chen (1956-2020) who was the main inspiration and guidance in her research in this area

Received  July 2020 Revised  December 2020 Published  February 2021

Fund Project: The first author is supported by the Taiwan MOST grant 107-2115-M-160-001-MY2. The second author is grateful for the financial support provided in part by the US NSF grant DMS-1818842

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.

Citation: Hsueh-Chen Lee, Hyesuk Lee. An a posteriori error estimator based on least-squares finite element solutions for viscoelastic fluid flows. Electronic Research Archive, doi: 10.3934/era.2021012
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Reduction of the functional norm $\delta g^{1/2}_{m}$ on three meshes for various weights $W_2 = 10^{m}$ ($2 \le m \le 6$)
$L^{2}$ errors of ${\boldsymbol \tau}$, ${\boldsymbol \sigma}$, ${p}$, $H^{1}$ error in ${\bf u}$, and $g^{1/2}$ functional norm of the LS solutions for (a) $W_2 = 10^{2}$ and (b) $W_2 = 10^{6}$
The slot channel with $d = h = l = 1$
$\Delta g^{1/2}_{(k)}/\Delta N_{k} = |g^{1/2}_{(k)}-g^{1/2}_{(k-1)}|/|N_{k}-N_{k-1}|$ versus the number of elements $N_{k}$ for $k =$1, 2, and 3, where $g^{1/2}_{(k)}$ denotes the functional value by $N_{k}$
Mesh G with various number of elements $N_k$ in refinement steps
The LS solutions using $W_2 = 10^{6}$ on the uniformly refined Mesh U with 6144 elements. Reduction of functional norm $\delta g^{1/2}_{m}$ and $u(2.5,0.5)$ for various weights $W_2 = 10^{m}$ ($2 \le m \le 6)$
Streamlines of LS solutions on the refined Mesh U with 6144 elements using (a) $W_2 = 10^{4}$ and (b) $W_2 = 10^{6}$
LS solutions at $Re = 1$, $We = 0.1$, and $\epsilon = 0.5$ using $W_2 = 10^{6}$. (a) Grid independence results for $g^{1/2}$. (b) Convergence of $g^{1/2}$ by the LS and ALS methods
Plots of $g^{1/2}_e$ on (a) the initial Mesh T with 1536 elements, (b) the uniformly refined Mesh U with 24576 elements, and (c) the adaptive Mesh G with 12993 elements
Plots of $g^{1/2}_e$ on (a) the initial Mesh S with 384 elements and (b) the adaptive Mesh H with 18320 elements at the 5-th refinement step
Streamlines of the flow over a slot by the ALS method for (a) the exponential PTT ($a = 1$), (b) the linear PTT ($a = 0$), and (c) the Oldroyd-B ($\epsilon = 0$) models at $We = 1$
Viscoelastic fluid models for $(a,\ We,\ \beta,\ \epsilon)$
 Constitutive equation $a$ $We$ $\beta$ $\epsilon$ Newtonian 0 0 $\beta$ 0 Upper Convected Maxwell 0 $We$ 0 0 Oldroyd-B 0 $We$ $\beta$ 0 Linear PTT 0 $We$ $\beta$ $\epsilon$ Exponential PTT 1 $We$ $\beta$ $\epsilon$
 Constitutive equation $a$ $We$ $\beta$ $\epsilon$ Newtonian 0 0 $\beta$ 0 Upper Convected Maxwell 0 $We$ 0 0 Oldroyd-B 0 $We$ $\beta$ 0 Linear PTT 0 $We$ $\beta$ $\epsilon$ Exponential PTT 1 $We$ $\beta$ $\epsilon$
Meshes considered for $(We,\ \epsilon,\ Re)$ = $(0.1,\ 0.5,\ 1)$
 Mesh Type Method $N_k ^{a}$ $S ^{b}$ $k ^{c}$ Mesh S Initial uniform grids LS 348 4 – Mesh T Initial uniform grids LS 1536 4 – Mesh U Uniform refined grids with Mesh T LS 24576 5 2 Mesh G Adaptive grids by ${\boldsymbol g^{1/2}}$ with Mesh T ALS 12993 4 3 Mesh H Adaptive grids by ${\boldsymbol g^{1/2}}$ with Mesh S ALS 18320 4 5 aNk represents the number of elements at the k refinement step. b S is the number of Newton steps for convergence. c k is the number of mesh refinements.
 Mesh Type Method $N_k ^{a}$ $S ^{b}$ $k ^{c}$ Mesh S Initial uniform grids LS 348 4 – Mesh T Initial uniform grids LS 1536 4 – Mesh U Uniform refined grids with Mesh T LS 24576 5 2 Mesh G Adaptive grids by ${\boldsymbol g^{1/2}}$ with Mesh T ALS 12993 4 3 Mesh H Adaptive grids by ${\boldsymbol g^{1/2}}$ with Mesh S ALS 18320 4 5 aNk represents the number of elements at the k refinement step. b S is the number of Newton steps for convergence. c k is the number of mesh refinements.
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