Top PDF A posteriori least-squares finite element error analysis for the Navier-Stokes equations

The a posteriori error analysis presented in this article shows that the error indicator is perfectly reliable for the guidance of mesh refinement at least for the Stokes problems and is very effective for the Navier–Stokes problem. And the error estimator is also highly reliable for feedback error control in self-adaptive automatic computations. The implementation of the residual estimator is very simple. The error indicators can be computed strictly within each element without using any information from neighboring elements because they do not involve jump conditions across element boundaries and local boundary conditions. Therefore, together with the symmetric property of the algebraic system in a neighborhood of a solution (Bochev and Gunzburger, 1993), the adaptive procedure of least squares computations for the Navier–Stokes equations can be completely parallel on an element-by-element basis if a conjugate gradient solver is used (Jiang and Carey, 1987). For more numerical results of adaptive LSFE computations, we also refer to Hsieh et al. (1999).

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 e€ective 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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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–

The remainder of the article is organized as follows. In Section II, we propose a new stress- pressure-displacement formulation for the elasticity equations. This is then decomposed into two subsystems, the stress-pressure system and the displacement system, with respective appropriate boundary conditions. In Section III, a two-stage least-squares finite element procedure is given, as well as its fundamental properties. In Section IV, a priori estimates for the stress-pressure system are derived. In Section V, error analysis is presented. In Section VI, the condition numbers of the resulting linear systems are estimated. Finally, in Section VII, some numerical experiments are examined to demonstrate this approach.

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 suciently 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.

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]

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.

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.

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.

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.

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]

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.

the results of the WENO2 scheme on different grid systems. A comparison of the calculated results of the experimental data and the Jiang et al. 29 results is shown in Table 1. Notice that the  ow is grid resolved and that the sharp trailing edge produces almost the same solutionsas that obtained with a blunt trailingedge. Computed skin-friction distribution from the upper surface of the RAE 2822 airfoil for the case just presented is compared with experimental data in Fig. 8. The skin-frictionvalues are referred to the boundary- layer edge dynamic pressure. Generally, the computed results are in good agreement with experiment, with exceptions near the leading edge, where the skin-frictionquantity is difŽ cult to deŽ ne, and near the trailing edge. The computed skin-friction coefŽ cients by Jiang et al. 29 do not agree well with available experimental data. Figure 9 shows the contours of constant Mach numbers. Figure 10 shows the convergence history. Again, the WENO scheme gives a good convergencerate. Figure 11 shows the convergenceof lift and drag of the Ž ne grid system of the WENO2 – Roe scheme.

© 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

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.

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.

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.

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.

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]