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

  • 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[J]. Electronic Research Archive, 2021, 29(4): 2755-2770. doi: 10.3934/era.2021012

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  • 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.



    The numerical simulation of viscoelastic fluid flows is challenging due to the strong coupling of governing equations and the large algebraic system with multiple dependent variables. Viscoelastic behavior of the fluid is often represented by a nonlinear constitutive model in the form of hyperbolic partial differential equations, which requires a stabilization technique for finite element approximations such as the streamline upwinding Petrov–Galerkin (SUPG) method and the discontinuous Galerkin method. In addition, finite element spaces for the velocity, pressure, and stress should satisfy the inf-sup condition if the standard mixed method is applied for numerical simulations. To overcome such difficulties, least-squares (LS) methods are frequently used to simulate viscoelastic flow problems [4,6,9,14] because they provide the flexibility of choosing finite element spaces and no additional stabilization is needed. In this work we expand on the LS method [14] by considering a nonlinear a posteriori error estimator of residual type for the exponential Phan-Thien-Tanner (PTT) viscoelastic fluid.

    The PTT model is a popular viscoelastic model in polymeric fluid simulations [17,16], where the extra-stress is written as a superposition of the polymeric and viscous stresses. The linear PTT model was proposed by Phan-Thien and Tanner in 1977, and Phan-Thien subsequently proposed the exponential PTT model in 1978. Ferrˊas et al. [7] reported that the exponential PTT model could behave more realistically in strong flows in which certain strain components grow exponentially in time. They also reported that the exponential model provided a better fit to experimental data.

    A posteriori error estimators play an important role in measuring the accuracy of numerical solutions and can be used to guide adaptive refinements. They are also effective and reliable for error control [1,4,15]. Previous works on the Stokes equations and the linear PTT model [11,14] show that an a posteriori error estimator serves as an indicator to adjust mass conservation weights on LS approaches to solve flows past a transverse slot. However, the numerical computations have been performed on uniform grids with more degrees of freedom. In [13], the results also show that grid effects of the LS method can not be reduced on some uniform refined grids, and geometric discontinuities cause corner singularities for flows in complex geometries.

    On the basis of these studies, we consider an a posteriori error estimator for the exponential PTT model to adjust mass conservation weights and to develop mesh refinement criteria on the LS approach. Furthermore, we estimate the coercivity and continuity for the homogeneous LS functional, which involves the sum of the linearized equation residuals measured in the L2-norm. The analysis of the linearized viscoelastic fluid using the LS method is extended from the proof presented in [14]. We propose an a posteriori error estimator of a residual type for the LS method as per considerations in the literature [8,11,15]. The convergence of the estimator is verified through convergence tests for a non-physical problem with a known solution, using properly adjusted weights for the LS functional. We extend the implementation to simulate the flow through a transverse slot using conforming piecewise polynomial elements for all unknowns. To capture the flow region and understand the flow feature, we modify the adaptive mesh algorithm in [10] using an a posteriori error estimator to refine grid points, and we compare numerical results by uniform and adaptive grids.

    The organization of this paper is as follows. Section 2 presents the exponential PTT flow model and the LS functional for the Newton linearized system. In Section 3 the least squares problem is defined and the coercivity and continuity properties of the LS functional are proved. Section 4 presents an a priori error estimate for the LS approximations and introduces an a posteriori error estimator. Section 5 provides numerical results for numerical examples, and finally conclusions follow in Section 6.

    The PTT model is one of widely used viscoelastic models in polymeric fluid simulations, where the extra-stress is written as a superposition of the polymeric stress σ and viscous stress τ [2]. The viscous stress, which is associated with a solvent in some applications, is Newtonian, i.e., τ=2ηsD, where ηs is a constant viscosity and D=0.5(u+uT) is the standard strain rate tensor with the velocity u. In the linearized form of the exponential PTT model, the polymer contribution to the stress obeys the following equation [2]

    σeϵληptr(σ)+λ(uσ(u)Tσσu)=2ηpD. (1)

    In (1) λ is a relaxation time and ηp is the polymeric contribution to the viscosity, ϵ is the extensibility parameter in the range of 0<ϵ<1. The exponential term eϵληptr(σ) can be expressed as

    eϵληptr(σ)=1+ϵληptr(σ)+12(ϵληp)2(tr(σ))2+O((ϵληp)3),

    hence,

    eϵληptr(σ)1+ϵληptr(σ)+a2(ϵληp)2(tr(σ))2. (2)

    The constitutive equation (1) with a=1 is often referred to as the exponential PTT model and the equation is reduces to the linear PTT model when a=0 [7].

    Consider an incompressible exponential PTT viscoelastic fluid flow in ΩR2 with the boundary Γ. Define the differential operators Gi for i=1,2,3,4 to describe the momentum equation, the mass conservation equation, the viscous constitutive equation, and the polymeric constitutive equation, respectively:

    G1U:=Re(uu)στ+p, (3)
    G2U:=u, (4)
    G3U:=τ2βD, (5)
    G4U:=We(uσ(u)Tσσ(u))+σ˜e(σ)2(1β)D, (6)

    where

    ˜e(σ):=1+ϵWe1βtr(σ)+a2(ϵWe1β)2(tr(σ))2.

    Collecting the constitutive equation (1) with the substitution (2) and by nondimensionalizaing, we have governing equations of the viscoelastic flow represented by the boundary value problem:

    GU=FinΩ, (7)
    u=0 onΓ, (8)

    where U:=(u,p,τ,σ), G:=(G1,G2,G3,G4), F:=(f,0,0,0).

    We assume that the scalar pressure p is fixed to p0 at the point x0 on Γ, i.e., p(x0)=p0, in order to ensure the uniqueness of pressure. Re>0 is the Reynolds number given by ReLcUcρ/η0, in which ηo=ηs+ηp is the zero-shear-rate viscosity, Lc and Uc are characteristic length and velocity, respectively. The parameter β=ηs/ηo[0,1] is the ratio of solvent viscosity to the total zero-shear-rate viscosity, which vanishes in the upper-convected Maxwell model. We is the Weissenberg number defined by WeλUc/Lc and f is the body force. In the case of We=0, the model is reduced to the Newtonian model, the Navier-Stokes equations. The Oldroyd-B model is obtained if ϵ=0 in (2). Different types of the fluid models are summarized in Table 1.

    Table 1.  Viscoelastic fluid models for (a, We, β, ϵ).
    Constitutive equation a We β ϵ
    Newtonian 0 0 β 0
    Upper Convected Maxwell 0 We 0 0
    Oldroyd-B 0 We β 0
    Linear PTT 0 We β ϵ
    Exponential PTT 1 We β ϵ

     | Show Table
    DownLoad: CSV

    To define the LS functional, we consider Newton linearization of the nonlinear equations (7)-(8) about known approximation un, σn of the velocity and the polymeric stress tensor, respectively. The linearized system of the PTT model may now be written as

    LU=F inΩ, (9)
    u=0 onΓ, (10)

    where L:=(L1,L2,L3,L4) and F:=(f1,f2,f3,f4), in which

    L1U:=Re(unu+uun)στ+p, (11)
    L2U:=u, (12)
    L3U:=τ2βD, (13)
    L4U:=L4,1U+L4,2U+L4,3U+L4,4U, (14)

    and

    A(u,σ)=(u)Tσ+σ(u),L4,1U:=σ2(1β)D,L4,2U:=We(uσn+unσA(un,σ)A(u,σn)),L4,3U:=(ϵWe1β)(tr(σn)σ+tr(σ)σn),L4,4U:=a2(ϵWe1β)2[(tr(σn)tr(σn))σ+2(tr(σn)tr(σ))σn],

    and

    f1:=f+Re(unun),f2:=0f3:=0f4:=We(unσn+A(un,σn))+(ϵWe1β)(tr(σn)σn)+a2(ϵWe1β)2(tr(σn)tr(σn))σn.

    The LS functional for (9)-(10) is then defined by

    J(U;F)=4j=1ΩWj(LjUfj)2dΩ. (15)

    The weights of positive constants W1=1, W2=10m for m=1 to 10, W3=1, and W4=(1+We+ϵ)2 are chosen on the basis of numerical tests under a wide range of parameters (0We1, 1Re10, 0ϵ1) in [14]. It is also shown in the work that the number of Newton iterative steps in the LS method is reduced by W4. Note that the choice of weights is not dependent on Re because inertial effects in the viscoelastic fluid flows at low Re can be neglected.

    Remark 1. Since the system of governing equations (7)-(8) is nonlinear, the linear functional (15) is minimized on each Newton iteration with un, σn replaced by the solution of the previous Newton step.

    We consider the product space

    Φ:=V×Q×Σs×Σ:=H1(Ω)2×L2(Ω)×L2(Ω)2×L2(Ω)2.

    The least-squares minimization problem for the solution of system (9)-(10) is to choose UΦ such that

    J(U;F)J(V;F)  VΦ, (16)

    i.e., seek U such that J(U;F) is minimized over Φ.

    To prove the coercivity and continuity of J, we first define

    J0:=3j=1WiLjU20+W4L4,1U+L4,2U+L4,3U20. (17)

    Here, the norm 0 is associated with the inner products (,)0, Ω. The functional J0 can be used for the linear PTT, i.e., J0=J(U;0) when a=0 (or L4,4U=0 in (14)), and the a priori estimate for J0 is presented in [14]. Based on the result in [14] and (14), we establish an a priori estimate for the LS functional (15) in Theorem 3.1.

    Theorem 3.1. Suppose the known approximations un, σn are uniformly bounded satisfying un=0, and

    M:=max{un, un,σn, σn}<. (18)

    Then, for any U=(u,p,τ,σ)Φ, there are positive constants, c0 and c1, which depend on Ω, β, We, α, and M, such that

    c0(u21+p20+τ20+σ20)J(U;0)

    and

    J(U;0)c1(Re2u21+p21+τ21+σ21)

    if M is sufficiently small.

    Proof. For a lower bound of J(U;0) we first consider the following estimate for L4,4U.

    L4,4U0=a2(ϵWe1β)2((tr(σn)tr(σn))σ0+2(tr(σn)tr(σ))σn0)3a(ϵWe1β)2σn20σ03a(ϵWe1β)2M2σ0, (19)

    and using the inequality a+b20(1/2)a20b20, we have

    J(U;0)12J0W4L4,4U2012J03aW4(ϵWe1β)2M2σ0. (20)

    It was proved in [14] that there exists a constant ¯C0>0 satisfying

    J0¯C0(u21+p20+τ20+σ20)

    if M is sufficiently small. Therefore, if M2¯C012aW4(1βϵWe)2,

    J(U;0)¯C02(u21+p20+τ20)+(¯C02¯C04)σ20c0(u21+p20+τ20+σ20), (21)

    where c0=¯C04. Finally, the upper bound follows naturally from the triangle inequality.

    For the finite element approximation of (7)-(8), we assume that the domain Ω is a polygon and that Th is a collection of finite elements such that Ω=eThe with h=max{diam(e):eTh}. Assume that the triangulation Th is regular and satisfies the assumption of inverse estimates [12]. The grid size is defined as h=2A/N, where A is the area of the domain and N is the number of elements in Th. Let Pr(e) denote the space of polynomials of degree less than or equal to r on element e. Define finite element spaces for the approximate of (u, p, τ, σ) by

    Vh={vhvhV(C0(Ω))2, vhe Pr+1(e)2 eTh},Qh={qhqhQC0(Ω), qhe Pr+1(e) eTh},Σhs={ζhζh Σs(C0(Ω))2×2, ζhe Pr+1(e)2×2 eTh},Σh={ςhςh Σ(C0(Ω))2×2, ςhe Pr+1(e)2×2 eTh}.

    Let Φh:=Vh×Qh×Σhs×Σh be finite element subspaces of Φ with the following approximation prosperities. Let Sh={uC0(Ω):u|ePr+1(e) eTh} admit the property

    infuhShuuhlChmum+luHm+l(Ω), (22)

    for mr+1 and l=0, 1.

    The discrete minimization problem for (16) is to choose UhΦh such that

    J(Uh;F)=infVhΦhJ(Vh;F), (23)

    where Uh=(uh,ph,τh,σh) and Vh=(vh,qh,ζh,ςh). Since Φh is a finite element subspace of Φ. Then the solution is equivalently to find UhΦ such that

    B(Uh,Vh)=F(Vh),  VhΦ, (24)

    where

    B(Uh,Vh):=(LUh,LVh), (25)
    F(Vh):=(F,LVh). (26)

    The following error estimate for the solution of (23) is obtained by the standard way, using the coercivity and continuity properties of the functional J in Theorem 3.1.

    Theorem 4.1. Consider approximating the solution to (9)-(10), where the known functions un, σn are sufficiently close to u, σ, respectively, and satisfy the condition (18). Assume that UΦ(Hm+1(Ω))2×Hm+1(Ω)×(Hm+1(Ω))2×2×(Hm+1(Ω))2×2 is the solution to (16) and UhΦh is the unique approximation solution to (23) for a small M. Then there is a positive constant C which is independent of h such that

    uuh1Chm(τm+1+σm+1+pm+1+Reum+1), (27)
    pph0Chm(τm+1+σm+1+pm+1+Reum+1), (28)
    ττh0Chm(τm+1+σm+1+pm+1+Reum+1), (29)
    σσh0Chm(τm+1+σm+1+pm+1+Reum+1), (30)

    for mr+1.

    Proof. By Theorem 3.1 and the approximation properties in (22), we have

    uuh1(uuh21+pph21+ττh21+σσh21)1/2infVhΦhc1c0(Re2uvh21+pqh21+τζh21+σςh21)1/2Chm(τm+1+σm+1+pm+1+Reum+1). (31)

    By the same approach, we can obtain the desired estimates for pph0, ττh0, and σσh0, respectively.

    Note that we obtain the error bounds O(h) in the L2-norm for τ, σ, and p, O(h) in the H1-norm for u if continuous piecewise linear polynomials are used to approximate all unknowns functions. Hence, we have the optimal convergence rate of the velocity in the H1-norm and suboptimal convergence rates of the stress and pressure in the L2-norm.

    For the residual of the first-order system (7)-(8), consider a nonlinear a posteriori error estimator of the following form:

    g=eThge,wherege=4j=1GjUhF20,e (32)

    and Uh=(uh,ph,τh,σh) is the finite element solution of (7)-(8) obtained by solving (24) iteratively by Newton's method. In numerical experiments the a posteriori error estimator g will be used to adjust the mass conservation weight W2 in the LS functional (15). Also the convergence behavior of g1/2 with respect to h will be numerically tested in the next section.

    Theorems 3.1 and 4.1 require the small M condition, where M is a bound of the velocity and the stress. A viscoelastic fluid flow with a small M is a low-velocity flow, hence, we will consider low Re flows only for numerical tests.

    We consider two numerical examples. The first problem is chosen for convergence tests with the known exact solution in the unit square domain, and the second is a slot flow problem. All variables are approximated by P1 polynomials, and the initial un and σn for Newton linearization are set to zero in all computations.

    Consider the flow in a planar channel on the domain [0,1]×[0,1] with the line of symmetry along y=0. Let u=(u,v) be specified along the boundary except on the axis of symmetry, and σ be specified on the inflow boundary where un<0. The pressure p=1 is set at the point (1, 0), and v and σxy vanish on the axis of symmetry. The exact solution is chosen as u=(1y4, 0)T, p=x2, σxx=2We(1β)˙γ2, σxy=(1β)˙γ, and σyy=0, where σ represents the analytical solution for the linear PTT model of a steady-state shear flow [3]. The source terms in (7) are chosen appropriately for the exact solution. The modeling parameters are selected as (a, We, ϵ, Re, β) = (1, 0.2, 0.5, 1, 1/9).

    First, in order to appropriately adjust the weight W2=10m in (15), three uniform directional triangular meshes with 512, 2048, and 8192 elements are initially used. We iterate over m for m=2,3,,10, to investigate convergence of the functional g1/2 in (32). Convergence of the iteration scheme with W2=10m is declared when the relative norm of residual in the nonlinear functional between two consecutive iterations, δg1/2m:=g1/2m+1g1/2m/g1/2m+1, is less than 104. See Figure 1 for δg1/2m by different m values. We observe that convergence is achieved at m=6, therefore, the mass conservation weight W2=106 is used in the LS formulation for all computations in the first example.

    Figure 1.  Reduction of the functional norm δg1/2m on three meshes for various weights W2=10m (2m6).

    Figure 2 shows the errors of LS solutions with W2=102 and W2=106, where for the case of W2=106 we observe the consistent rates with the error estimate in Section 4; the rate for the velocity is optimal, O(h) in the H1-norm, and the rates for viscous and polymeric stresses, and pressure are suboptimal, O(h) in the L2-norm. Figure 2 shows that the convergent rates of LS solutions are higher for the larger W2(=106). These results also show the optimal convergence of g1/2 at O(h).

    Figure 2.  L2 errors of τ, σ, p, H1 error in u, and g1/2 functional norm of the LS solutions for (a) W2=102 and (b) W2=106.

    Consider the PTT flow past a slot in a channel for 2.5x2.5 with the contraction occurring at x=0. See Figure 3. Let u=(u,v) and σ be specified on the inflow boundaries, and u=0 on the wall boundaries. Pressure p and v are set to zero at the outflow boundary. The forcing function f in (15) is set to 0. The system of model equations (7)-(8) is simulated with the parameter values (a, We, ϵ, Re, β) = (1, 0.1, 0.5, 1, 1/9).

    Figure 3.  The slot channel with d=h=l=1.

    We consider the LS method (15) with the adaptively refined algorithms, which we name it the ALS method. This refinement technique is similar to the approach used for the Carreau model in [10]. For the mesh refinement strategy, we apply the grading function ϕe on elements e of a Mesh T given by ϕe=qe, where the scalar function q is calculated based on the a posteriori error estimator of LS solutions, i.e, q=g1/2e. Then, as in [10], the mesh redistribution function f(ϕe) is defined by

    f(ϕe)=|e|max(ϕeϕemin)Δ|e|Δϕe,

    where |e|max=max{|e|:eT}, Δ|e|=|e|max|e|min and Δϕe=ϕemaxϕemin. If e T satisfies f(ϕe)|e|, then e is subdivided. We employ the a posteriori error estimator for mesh refinement criteria: Δg1/2(k)/ΔNk=|g1/2(k)g1/2(k1)|/|NkNk1| where Nk represents the number of elements at the k-th refinement step and g1/2(k) denotes the functional value by Mesh Tk. Figure 4 shows that a mesh convergence for Nk is confirmed when Δg1/2(k)/ΔNk <2×105, which yields convergence of Mesh U at k=2 with N2=25000 and Mesh G at k=3 with N3=12993, respectively. Figure 5 shows Mesh G in the k-th refinement step for k=1, 2, and 3 by the ALS method starting from the initial mesh T. The results show these adaptive meshes are highly refined near the corner. More details about meshes are given in Table 2.

    Figure 4.  Δg1/2(k)/ΔNk=|g1/2(k)g1/2(k1)|/|NkNk1| versus the number of elements Nk for k=1, 2, and 3, where g1/2(k) denotes the functional value by Nk.
    Figure 5.  Mesh G with various number of elements Nk in refinement steps.
    Table 2.  Meshes considered for (We, ϵ, Re) = (0.1, 0.5, 1).
    Mesh Type Method Nak Sb kc
    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 g1/2 with Mesh T ALS 12993 4 3
    Mesh H Adaptive grids by g1/2 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.

     | Show Table
    DownLoad: CSV

    The optimal weight W2 of the form W2=10m is chosen in a similar manner to the first example; we iterate over m for 2m6 for the functional g1/2m in (32) on the uniformly refined Mesh T with 6144 elements. As shown in Figure 6, δg1/2m=|g1/2m+1g1/2m|/g1/2m+1 is less than 104 at m=6, therefore, W2=106 is chosen as the optimal weight for the LS formulation. Figure 6 also shows u(2.5, 0.5), the horizontal velocity u at (2.5, 0.5), converges to 1.546 for W2104. Figure 7 shows streamlines of the LS solutions using W2=104 and W2=106, respectively. The figure shows that satisfactory solutions can be obtained by a sufficiently large weight, W2=106, as shown in [11].

    Figure 6.  The LS solutions using W2=106 on the uniformly refined Mesh U with 6144 elements. Reduction of functional norm δg1/2m and u(2.5,0.5) for various weights W2=10m (2m6).
    Figure 7.  Streamlines of LS solutions on the refined Mesh U with 6144 elements using (a) W2=104 and (b) W2=106.

    Grid independence results of g1/2 and the convergence of g1/2 by LS and ALS methods are presented in Figure 8(a) and (b), respectively. The figure shows that both LS and ALS methods yield the same convergent results of g1/2, O(h). However, using the ALS method, the number of unknowns can be reduced from 111648 to 59551, therefore, the ALS method on Mesh G is more efficient than the LS method on Mesh U.

    Figure 8.  LS solutions at Re=1, We=0.1, and ϵ=0.5 using W2=106. (a) Grid independence results for g1/2. (b) Convergence of g1/2 by the LS and ALS methods.

    Furthermore, in Figure 9, we present contours of g1/2e on uniform Meshes T and U by the LS method and adaptive Mesh G by the ALS method. Figures 9(b) and 9(c) show similar contours of g1/2e on Meshes U and G, respectively, and as shown in Figure 9(a), g1/2e near the corner on Mesh T can be reduced using Meshes U and G. Thus, Meshes U and G reduce the residual error of the system generated by corner singularity. Using the ALS scheme, the number of elements can be reduced from 24576 (Mesh U) to 12993 (Mesh G), thus, the ALS method on Mesh G is more efficient than the LS method on Mesh U. Figure 10 presents contours of g1/2e on the coarse uniform Mesh S and the adaptive Mesh H, respectively. The results indicate that g1/2e on Mesh S can be reduced by Mesh H. Note that the contours of g1/2e on Meshes G and H in Figures 9(c) and 10, respectively, are almost in agreement. For the initial Mesh T, the number of elements and the refinement steps can be reduced from 18320 (Mesh H) to 12993 (Mesh G) and from 5 (Mesh H) to 3 (Mesh G), respectively. Thus, Mesh T as the initial Mesh for the ALS method outperforms the initial Mesh S.

    Figure 9.  Plots of g1/2e 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.
    Figure 10.  Plots of g1/2e on (a) the initial Mesh S with 384 elements and (b) the adaptive Mesh H with 18320 elements at the 5-th refinement step.

    Finally, we compare streamlines of the ALS solutions for the exponential PTT (a=1), linear PTT (a=0), and the Oldroyd-B (ϵ=0) models at We=1. Figure 11 shows that vortices for We=1 fluids. The vortices of the fluids move toward the reentrant corners and are consistent with those obtained on uniform grids in [14]. The result shows that the shear-thinning behavior of the exponential PTT flow seems to be more obvious than the linear PTT or Oldroyd-B flows.

    Figure 11.  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 (ϵ=0) models at We=1.

    We considered a nonlinear a posteriori error estimator based on a first-order LS method for the exponential PTT viscoelastic model of strong flows. The LS method was developed by assigning proper weights to the terms of the LS functional, in which the mass conservation weights are obtained by the a posteriori error estimator. We also proposed an adaptive LS (ALS) for the exponential PTT model, where the a posteriori error estimator of the LS solution is used for adaptive mesh refinements. The proposed approach resolves some difficulties related to the presence of corner singularities in this second example. In addition, the ALS approach with adaptive mesh refinements improves computational efficiency.



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