First-Order System Least-Squares for the Oseen Equations
Total Page：16
File Type：pdf, Size：1020Kb
FIRST-ORDER SYSTEM LEAST-SQUARES FOR THE OSEEN EQUATIONS SANG DONG KIM¤, CHANG-OCK LEEy , THOMAS A. MANTEUFFELz , STEPHEN F. MCCORMICKx , AND OLIVER ROHRLEÄ { Abstract. Following earlier work for Stokes equations, a least-squares functional is developed for two- and three-dimensional Oseen equations. By introducing a velocity flux variable and associated curl and trace equations, ellipticity is established in an appropriate product norm. The form of Oseen equations examined here is obtained by linearizing the incompressible Navier-Stokes equations. An algorithm is presented for approximately solving steady state, incompressible Navier-Stokes equations with a Nested Iteration-Newton-FOSLS-AMG iterative scheme, which involves solving a sequence of Oseen equations. Some numerical results for Kovasznay flow are provided. Key words. least-squares discretization, Oseen equations, nested iteration AMS(MOS) subject classi¯cations. 65F10, 65F30 1. Introduction. Over the last decade, many successful theories were developed for a ¯rst- order system formulation of Stokes and Navier-Stokes equations (see [1, 2, 3, 4, 5, 6, 7]). The ¯rst- order system introduced in [7] for Stokes equations involved velocity, u, a new dependent variable, U = rut, refer to as velocity flux, and pressure, p. The least-squares functional using L2 and H¡1 norms weighted appropriately by the viscosity constant, ¹, was shown to be elliptic in the sense that 2 2 2 2 2 its homogeneous form is equivalent to squared product norm ¹ kUk +¹ kuk1 +kpk : Furthermore, with the extended ¯rst-order system, which includes equations involving trace and curl related to velocity flux, it was shown that the functionals using an L2 norm weighted by ¹ is elliptic in the sense 2 2 2 2 2 of squared product norm ¹ kUk1 +¹ kuk1 +kpk1 (see [7]). This equivalence of the ¯rst-order system least-squares method enables us to use standard ¯nite element spaces to discretize the equations and multigrid methods to solve the resulting linear systems with respective optimal approximation and solver properties (see, for example, [8]). The purpose of this paper is to apply this methodology to the steady-state, incompressible Navier-Stokes equations with a linearized convection term, which yields the Oseen equations. A method for linearization of the Navier-Stokes equations is to introduce a known function, ¯, con- sidered as an approximation of velocity, u, in the convection terms. Thus, (rut)tu is replaced by (r(ut + ¯t))t(u + ¯), which is linearized as (rut)t¯ + (r¯t)tu + (r¯t)t¯. As in [7], we begin by reformulating the Oseen equations in two or three dimensions as a ¯rst-order system in terms of a velocity flux variable, and then apply least-squares principles to this system using L2 and H¡1 norms, which produces a functional we call G1 (see (3.1)). This system can be further extended to a ¯rst-order system with equations involving the curl and grad trace of the velocity flux variable. 2 The associated least-squares functional, formed by summing L norms of each equation, we call G2 ¤Department of Mathematics Education, Kyungpook National University, Taegu, Korea 702-701 ([email protected]). This work was sponsored by the Korea Research Foundation under grant KRF- 2002-070-C00014. yDivision of Applied Mathematics, KAIST, Daejeon, 305-701, Korea ([email protected]). This work was sponsored by KOSEF R01-2000-00008. zDepartment of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309- 0526 ([email protected]). This work was sponsored by the Department of Energy under grant numbers DE-FC02-01ER25479 and DE-FG02-03ER25574, Lawrence Livermore National Laboratory under contract number B541045, Sandia National Laboratory under contract number 15268, and the National Science Foundation under grant numbers DMS-0410318.. xDepartment of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309- 0526 ([email protected]). This work was sponsored by the Department of Energy under grant numbers DE-FC02- 01ER25479 and DE-FG02-03ER25574, Lawrence Livermore National Laboratory under contract number B541045, Sandia National Laboratory under contract number 15268, and the National Science Foundation under grant numbers DMS-0410318. {Bioengineering Institute, The University of Auckland, Private Bag 92019, Auckland 1, New Zealand ([email protected]). This work was sponsored by the Department of Energy under grant numbers DE-FC02- 01ER25479 and DE-FG02-03ER25574, Lawrence Livermore National Laboratory under contract number B533502, Sandia National Laboratory under contract number 15268, and the National Science Foundation under VIGRE grant number DMS-9810751. 1 (see (3.2)). We assume that the Oseen equations have a unique solution, which provides the desired 1 2 H and H regularities (see (3.5) and (3.12)). We are, thus, able to show that G1 is equivalent to 2 1 1 the squared L and H norms and that G2 is equivalent to the squared product H norm. It should be noted that the constants involved in the equivalences we develop in this paper are naturally 1 dependent on the domain, , the Reynolds number, ¸ = ¹ , and the approximate velocity, ¯. Incompressible Navier-Stokes equations can be approximately solved by initial triangulation of the physical domain using a ¯nite element grid that is as coarse as possible, but ¯ne enough to roughly resolve the domain. A Newton iteration is employed, which involves solving a sequence of Oseen equations, each of which is discretized using a FOSLS formulation. The resulting discrete equations are solved using an algebraic multigrid method (AMG). Once the Navier-Stokes equations on the initial grid are su±ciently resolved, the solution is used as an initial guess for a re¯ned grid. This Nested-Iteration (NI) algorithm is repeated through a sequence of successively ¯ner grids until su±cient resolution of the Navier-Stokes equations is achieved. Numerical results presented in section 6 show that this approach is highly e®ective. Computation on the ¯nest grid is minimized, which is advantageous because work on the coarse grids is relatively inexpensive. After the ¯rst few grids, the solution on one grid is in the basin of attraction of the next ¯ner grid, where only one Newton step, that is, the solution of one Oseen system, is required. The approach here is similar to the NI-Newton-FOSLS-AMG scheme described in [9] and applied to elliptic grid generation in [10]. It is well known that the basin of attraction for such methods can be very small. To get within the basin of attraction, one often has to use on coarser levels more expensive and problem- speci¯c methods such as the Projection Multilevel (PML) method proposed in [11], which treats the nonlinearity of the Navier-Stokes equations directly and does not appeal to any linearization step. This method tends to be more robust and has typically a larger domain of convergence, but is in its current form limited, for example, by its set-up cost and choice of smoother. A more signi¯cant advantage of global linearization methods, e.g., the NI-Newton-FOSLS-AMG scheme, is that they rely on the solution of large linear systems of equations and, therefore, can draw upon an extensive repertoire on algorithms and knowledge, in particular, since substantial multigrid research is directed on developing robust, fast, and e±cient linear solvers. In section 2, we provide the ¯rst-order system formulation with de¯nitions, notation, and some preliminaries. In section 3, ellipticity and continuity for the functionals are shown. In section 4, H1 and H2 regularity is proved under the assumption of uniqueness of the solution of the Oseen equations. The NI-Newton-FOSLS-AMG algorithm is described in section 5. In section 6, numerical results for Kovasznay flow are presented. 2. The Oseen equations, its ¯rst-order system formulation, and other preliminaries. Let be a bounded, connected domain in <n(n = 2; 3) with Lipschitz continuous (C0;1) boundary, @. Consider the stationary, incompressible Oseen equations with Dirichlet boundary conditions for t velocity, u = (u1; : : : ; un) , and the usual integral condition for pressure, p, as follows: 8 < ¡¹¢ u + (rut)t¯ + (r¯t)tu + rp = f; in ; (2.1) r ¢ u = g; in ; : u = b; on @; where symbols ¢,r, and r¢ stand for the Laplacian, gradient, and divergence operators, respectively t 1 (¢u is the vector of components ¢ ui); ¯ = (¯1; : : : ; ¯n) is a given C vector function; ¹ is a viscosity constant; f is a given vector function; g is a given scalar function; and b is the value of velocity on the boundary. We further assume that Z p d = 0: For convenience, we adopt the notation introduced in [7]. A new independent variable related to 2 the n -vector function of gradients of the displacement vectors, ui, i = 1; : : : ; n, is introduced below. It is convenient to view the original n-vector functions as column vectors and the new n2-vector t functions as either block column vectors or matrices. The velocity variable, u = (u1; : : : ; un) , is a 2 t column vector with scalar components ui, so that the gradient, ru , is a matrix with columns rui. For a function, U, with n vector components Ui, we write t U = ru = (U1; U2;:::; Un) = (Uij )n£n; which is a matrix with entries Uij = @uj=@xi, 1 · i; j · n: We next de¯ne the trace operator, tr, as Xn trU = Uii; i=1 the divergence operator, r¢, as t t (r ¢ U) = (r ¢ U1; r ¢ U2;:::; r ¢ Un) ; and the curl operator, r£, as r£U = (r£U1;:::; r£Un): Note that, in the two-dimensional case, n = 2, the curl of u is naturally de¯ned by means of the scalar function µ ¶ u1 r£ = @1u2 ¡ @2u1: u2 We also de¯ne the tangential operator, n£, componentwise as n £ U = (n £ U1;:::; n £ Un): The inner products and norms on the block column vector functions are de¯ned in the natural componentwise way; for example, Xn Xn 2 2 2 kUk = kUik = kUijk : i=1 i;j=1 We use standard notation and de¯nitions for the Sobolev spaces Hs()n, associated inner products 0 n <¢; ¢>s, respective norms, k ¢ ks, and semi-norms, j ¢ js, for s ¸ 0.