Abstract
**A** residual type **a** **posteriori** **error** estimator is presented **for** **the** **least** **squares** ®nite **element** method. **The** estimator is proved to equal **the** exact **error** in **a** norm induced by some **least** **squares** functional. **The** **error** indicator of each **element** is equal to **the** exact **error** norm restricted to **the** **element** as well. In other words, **the** estimator is perfectly eective and reliable **for** **error** control and **for** adaptive mesh re®nement. **The** exactness property requires virtually no assumptions on **the** regularity of **the** solution and on **the**

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Taylor & Francis makes every effort to ensure **the** accuracy of all **the** information (**the** “Content”) contained in **the** publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to **the** accuracy, completeness, or suitability **for** any purpose of **the** Content. Any opinions and views expressed in this publication are **the** opinions and views of **the** authors, and are not **the** views of or endorsed by Taylor & Francis. **The** accuracy of **the** Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable **for** any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of **the** use of **the** Content.

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6. Conclusions
In this paper, we propose **a** new Cartesian ﬁnite difference method **for** solving **a** coupled system of **a** ﬂuid ﬂow modelled by **the** **Navier**–**Stokes** **equations** and **a** porous media modelled by **the** Darcy law, respectively. Numerical results show that both **the** com- puted pressure and velocity are second order accurate in **the** max- imum norm **for** **the** problems with known analytic solutions. **The** new method can also guarantee **the** equivalence of **the** solutions between **the** original and **the** transformed problem under certain regularity assumptions though **the** sensitivity **analysis** of **the** new system is still an open question. **The** new approach has also been applied to **the** **Stokes** and Darcy coupling, which outperforms **the** method proposed in [27] in almost all **the** categories. **For** **Navier**–

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b Department of Statistics, Ming Chuan University, Taipei 11120, Taiwan, ROC
c Department of Applied Mathematics, I-Shou University, Ta-Hsu, Kaohsiung 84008, Taiwan, ROC
Abstract
In this paper **a** **least**-**squares** ®nite **element** method **for** **the** Timoshenko beam problem is proposed and analyzed. **The** method is shown to be convergent and stable without requiring extra smoothness of **the** exact solutions. **For** suciently regular exact solutions, **the** method achieves optimal order of convergence in **the** H 1 -norm **for** all **the** unknowns (displacement, rotation, shear, moment), uniformly in **the** small parameter which is generally proportional to **the** ratio of thickness to length. Thus **the** locking phenomenon disappears as **the** parameter tends to zero. **A** sharp **a** **posteriori** **error** es- timator which is exact in **the** energy norm and equivalent in **the** H 1 -norm is also brie¯y discussed. Ó 2000 Published by Elsevier Science Inc. All rights reserved.

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These methods offer certain advantages such as that they need not satisfy the inf-sup condition which is required in the mixed finite element formulation, that a sin[r]

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PLEASE SCROLL DOWN **FOR** ARTICLE
Taylor & Francis makes every effort to ensure **the** accuracy of all **the** information (**the** “Content”) contained in **the** publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to **the** accuracy, completeness, or suitability **for** any purpose of **the** Content. Any opinions and views expressed in this publication are **the** opinions and views of **the** authors, and are not **the** views of or endorsed by Taylor & Francis. **The** accuracy of **the** Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable **for** any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of **the** use of **the** Content.

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ow problem in **a** disk domain is because it suces to have all aforementioned numerical dif-
culties occurring in higher-dimensional cylindrical and spherical geometries. Note that here we are not trying to compare or compete with **the** nite **element** or spectral methods **for** solving such kind of problems. We simply want to introduce some simple nite dierence treatments **for** **the** diculties such as **the** co-ordinate singularity and **the** restrictive CFL con- straint. Another main feature of our method is **the** fourth-order accuracy which is sucient **for** most of ow computations. **The** rest of **the** paper is organized as follows. In Section 2, we formulate **the** incompressible **Navier**–**Stokes** **equations** using **the** vorticity-stream function form in polar co-ordinates. **The** complete fourth-order numerical scheme **for** solving **the** **equations** is described in Section 3. **A** simple fourth-order fast direct solver **for** Poisson equation in **a** disk is reviewed in Section 4. **The** numerical results are given in Section 5 and followed by some conclusion.

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Compared to **the** cost of computing **the** approximate solution u h , **the** cost of computing **the** es- timated **error** ˜e is fractional since **the** complementary BE space S h c can be constructed using, **for** instance, only one or two shape functions on each **element**. Consequently, we only have one or two **equations** in **the** solution of **a** local problem. In particular, if only one shape function is used **for** S h c , our estimator is then equivalent to **the** previous residual **error** estimators; see **the** numerical ex- ample **for** **a** hypersingular integral equation presented in Section 4. Furthermore, since **the** estimated **error** is explicitly calculated there is no restriction on **the** choice of **the** norm used to measure **the** errors. In other words, whichever **the** norm appropriate **for** **the** approximate solution u h can also be used **for** **the** estimated **error** ˜e. This can be very useful in practice when **a** more
exible norm is needed **for** assessing **the** computed solution. We however only prove estimate (1.1) in **the** energy norm.

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4. Global uniqueness **for** **the** stationary **Navier**-**Stokes** **equations** In this section we consider **the** unique determination of **the** viscosity in an in- compressible fluid described by **the** stationary **Navier**-**Stokes** **equations**. In higher dimensions, this problem has been solved by Li and Wang in [11] using **the** lineariza- tion technique. Since their methods are independent of spatial dimensions, we could apply their ideas to show **the** uniqueness result of µ **for** **the** **Navier**-**Stokes** **equations** in **the** two dimensional case.

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A general framework of the theoretical analysis for the convergence and stability of the standard least squares finite element approximations to boundary value problems of fir[r]

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4 w (1 + cos(πx/2w)), if |x| < 2w,
0 , if |x| ≥ 2w. (4.31)
Although there are other discrete delta functions in **the** literature, **the** computational results are not much different **for** two- and three-dimensional problems. **The** most common choices of w is h, **the** spatial mesh size. As we can see, when w gets larger, **the** cost to spread **the** singular force to **the** grid points increases significantly. Our numerical tests **for** this and other examples show that **the** accuracy of **the** computed velocity remains pretty much **the** same **for** different choices of w. However, **the** pressure, if it is discontinuous, smears out more widely as w increases. Therefore, **the** best choice of w is h. **The** sixth column in Table II is **the** grid refinement **analysis** **for** **the** velocity which is clearly first-order accurate.

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© 2007 Elsevier Inc. All rights reserved.
1. Introduction
In this work we consider **the** unique determination of **the** viscosity in an incompressible fluid described by **the** stationary **Navier**–**Stokes** **equations**. Let Ω ⊂ R 3 be an open bounded domain with boundary ∂Ω ∈ C ∞ . Assume that Ω is filled with an incompressible fluid. Let u = (u 1 , u 2 , u 3 ) T be **the** velocity vector field satisfying **the** stationary **Navier**–**Stokes** system

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Department of Electronics Engineering and Institute of Electronics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan Received 25 May 1998; received in revised form 5 January 2000
Abstract
This paper proposes **a** new algorithm **for** designing "nite word length linear-phase FIR "lters. **The** new algorithm produces "nite-precision **least**-**squares** **error** (LSE) solutions with much reduced search time than **the** brute-force full search algorithm. It is di!erent from **the** full search algorithm that tries all possible combinations directly. **The** new algorithm utilizes geometric properties of **a** hyper-space to pinpoint potential solutions in **a** much more restricted way.

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Another problem is that ENO schemes are not effective on vector supercomputers because **the** stencil-choosing step involves heavy usage of logical statements which perform poorly on such machines. **The** WENO schemes introduced by Liu et al. [5] and extended by Jiang and Shu [4] can overcome these drawbacks while keeping **the** robustness and high-order accuracy of ENO schemes. **The** concept of WENO schemes is **the** following: instead of approximating **the** numerical flux using only one of **the** candidate stencils, one uses **a** convex combination of all **the** candidate stencils. Each of **the** candidate stencils is assigned **a** weight which determines **the** contribution of this stencil to **the** final approximation of **the** numerical flux. **The** weights are defined in such **a** way that in smooth regions **the** stencil approaches certain optimal weights to achieve **a** higher order of accuracy, while in regions near discontinuities, **the** stencils which contain **the** discontinuities are assigned **a** nearly zero weight. Thus **the** essentially nonoscillatory property is achieved by emulating ENO schemes around discontinuities and **a** higher order of accuracy is obtained by emulating upstream central schemes with **the** optimal weights away from **the** discontinuities. Both efficient ENO and weighted ENO schemes have been extensively tested and applied to **the** compressible Euler/**Navier**–**Stokes** **equations**.

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We also have tested **the** method by embedding **the** **Navier**–**Stokes** **equations** to **the** entire domain and selecting **the** nor- mal and tangential forces along **the** boundary as augmented variables as described in several papers in **the** literature. **The** resulting Schur-complement system is severely ill-conditioned, see Table 1 **for** an illustration. But our selection of **the** aug- mented variable ½ @u @n @ X ¼ q kþ1 results in **a** very well-conditioned system. Since we are only interested in **the** solution in **the** region of R n X , **the** other terms such as ðu r uÞ kþ1=2 and r p k1=2 can be treated as **a** forcing term in **the** elliptic **equations** that do not need to be extended. These quantities at irregular grid points where **the** boundary @ X cuts through **the** standard cen- tral 5-point stencil, can be approximated by **a** one-sided interpolation scheme.

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In Section 3, we $rst analyze **the** matrix properties of **the** resulting adaptive $nite **element** sys- tems **for** **the** Poisson equation, which then lead to **the** M-matrix properties **for** **the** semiconductor **equations**. Starting with **the** upper and lower solutions as initial guesses, it is shown in Section 4 that maximal and minimal sequences generated by Picard, Gauss–Seidel, and Jacobi iterations all converge monotonically from above and belowto **the** unique solution of **the** resulting nonlinear sys- tem. We then summarize in Section 5 our implementation procedures into two algorithms, namely, monotone-Gummel and adaptive algorithms which combine Gummel’s decoupling, monotone itera- tive, and adaptive methods. Section 6 represents **a** part of our extensive numerical experiments on various n-MOSFET device models to demonstrate **the** accuracy and ethe proposed methods. Moreover, numerical results of **the** Jacobi and Gauss–Seidel monotone iterations are also given to verify **the** theoretical results.

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On the other hand, an attempt to create a univer- sal conditions for the general system (1) and (2) in the context of LSFE approx- imation is very intractable if not impossi[r]

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