^{1}

^{*}

^{2}

In this paper, a least-squares finite element method for the upper-convected Maxell (UCM) fluid is proposed. We first linearize the constitutive and momentum equations and then apply a least-squares method to the linearized version of the viscoelastic UCM model. The
L
^{2}
least-squares functional involves the residuals of each equation multiplied by proper weights. The corresponding homogeneous functional is equivalent to a natural norm. The error estimates of the finite element solution are analyzed when the conforming piecewise polynomial elements are used for the unknowns.

In recent years, there has been an increased interest in the least-squares finite element method for the approximation of partial differential equations, see e.g. [

In the viscoelastic fluids of the differential type, the constitutive equations consist of an algebraic tensorial relationship between the stress tensor and the rate of deformation tensor. The upper-convected Maxwell fluid [

Assume that

where

For the upper-convected Maxwell model, the extra-stress tensor

where

and

To simplify our analysis, homogeneous boundary conditions are assumed on

Throughout the paper, we use the standard notation and definition for the Sobolev spaces

The spaces

where

We use the following approximation

to linearize the equations in (2.4). Moreover we assume that

We introduce the replacement rules

which result in the linearized system

where

The velocity

and let

Based on [

Now we show that the homogeneous least-squares functional of (3.4) is equivalent to the norm

Theorem 1. There exist positive constants

hold for any

Proof. The upper bound in (3.5) follows easily from the triangle inequality and (3.2). For the lower bound, we will show that

where

Using the Greenâ€™s formula and Cauchy-Schwarz inequality, we obtain

for any

we obtain

which implies that

Similarly, we have

By the arguments similar to Theorem 4.1 in [

Combining (3.7)-(3.9) yields (3.6).

From the inequality

Note that

and, using (3.10),

where

However the least squares functional

We assume that the domain

We assume that the partition

Let

where

with

The mesh dependent least squares functional is defined by the weighted sum in

where ^{h}: seek

The minimizer of (3.13) necessarily satisfies the Euler-Lagrange equation given by

where

and the double-dot product is defined as

Based on Theorem 1, we establish the ellipticity of the functional

Theorem 2. For any

for

Proof. The first inequality in (3.15) is straightforward from Theorem 1. To prove the upper bound, we assume that the spaces

and

From the triangle inequality, we obtain

This completes the proof of the theorem.

By virtue of Theorem 2 and the Lax-Milgram theorem, we establish the following theorem.

Theorem 3. For any

Now we derive error estimates for the least-squares finite element solution

Theorem 4. Assume that

for

Proof. From Theorem 2, we obtain the following bound

Combining the properties (3.11) and (3.12), we have

This completes the proof of the theorem.

In this paper, we have proposed and analyzed a weighted least-squares method for the approximate solution of the upper-convected Maxwell fluid. The weights in our least-squares functional involve mesh dependent weight and mass conservation constant. The homogenous functional is shown to be equivalent to a natural norm. A prior error estimate is given for the finite element solutions. An adaptive least-squares finite element method for this viscoelastic fluid model will be discussed in the future.

The authorsâ€™ work is supported by the National Science Foundation of China (No. 11271247) and the Natural Science Foundation of Hebei Province (No. G2013402063).