A COMMENT ON LEASTSQUARES FINITE ELEMENT METHODS WITH MINIMUM REGULARITY ASSUMPTIONS


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1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 10, Number 4, Pages c 2013 Institute for Scientific Computing and Information A COMMENT ON LEASTSQUARES FINITE ELEMENT METHODS WITH MINIMUM REGULARITY ASSUMPTIONS JAEUN KU Abstract. Leastsquares(LS) finite element methods are applied successfully to a wide range of problems arising from science and engineering. However, there are reservations to use LS methods for problems with low regularity solutions. In this paper, we consider LS methods for secondorder elliptic problems using the minimum regularity assumption, i.e. the solution only belongs to H 1 space. We provide a theoretical analysis showing that LS methods are competitive alternatives to mixed and standard Galerkin methods by establishing that LS solutions are bounded by the mixed and standard Galerkin solutions. Key words. leastsquares, finite element methods, Galerkin methods. 1. Introduction The purpose of this paper is to show that LS methods could compete favorably with mixed Galerkin methods under minimum regularity assumptions, i.e. the solution only belongs to H 1 space. We consider firstorder LS methods for secondorder elliptic problems proposed by Cai et. al [6] and Pehlivanov et. al [10]. They transformed the secondorder equations into a system of firstorder by introducing a new variable(flux) σ = A u. Leastsquares type methods applied to the system lead to a minimization problem and resulting algebraic equation involves a symmetric and positive definite matrix. The approximate spaces do not require the infsup condition and any conforming finite element spaces can be used as approximate spaces. While the LS methods are successfully applied to a wide range of problems in science and engineering, there are reservations to use LS methods for problems with low regularity solutions such as problems with discontinuous coefficients, problems on nonconvex domains etc. This is due to the fact that most of the error estimates concerning LS methods require high regularity solutions. In this paper, we consider errors of LS solutions, and mixed and standard Galerkin solutions under the assumption, u 1 C f 1, and establish σ σ h 0 + u u h 0 C( σ σ m h 0 + u u G h 0 ), where (u h,σ h ) is the LS solution for (u,σ = A u), and σ m h and ug h is the mixed and Galerkin solution for σ and u, respectively. From this estimate, we observe that LS solutions compete favorably with mixed and Galerkin solutions. For other results concerning error estimates with minimum regularity assumptions, we refer the reader to [12], where the error estimates for the RitzGalerkin methods are presented with the minimum regularity assumption considered here. For results concerning smooth problems, we refer to [1, 3] and references therein. Received by the editors September 24, Mathematics Subject Classification. 65N30, 65N
2 900 J. KU 2. Problem formulation Let Ω be bounded domain in R n,n = 2,3 with Ω = Γ D ΓN, where Γ D. Let n denote the unit outward normal vector to the boundary. We consider the following boundary value problem: (1) (A u) = f in Ω, with boundary conditions (2) u = 0 on Γ D and n A u = 0 on Γ N, where the symbols and stand for the divergence and gradient operators, respectively; f L 2 (Ω) and A = (a ij (x)) n i,j=1,x Ω. We shall assume that a ij L (Ω) and the matrix A is symmetric and uniformly positive definite, i.e., there exist positive constants α 0 > 0 and α 1 > 0 such that (3) α 0 ζ T ζ ζ T Aζ α 1 ζ T ζ, for all ζ R n and all x Ω. We assume that there exists a unique solution to (1). Let H s (Ω) denote the Sobolev space oforder s defined on Ω. The norm in H s (Ω) will be denoted by s. For s = 0,H s (Ω) coincides with L 2 (Ω). We shall use the spaces with norms V = {u H 1 (Ω) : u = 0 on Γ D }, W = {σ (L 2 (Ω)) n : σ L 2 (Ω) and n σ = 0 on Γ N }, u 2 1 = (u,u)+( u, u) and σ 2 H(div) = ( σ, σ)+(σ, σ). We assume the following a priori estimate for u satisfying (1): there exists a positive constant C independent of f satisfying (4) u 1 C f 1. Here, f 1 is defined in the standard way by v 1 = sup φ V (v,φ) φ 1. As noted in [12], it is not known in general whether u H 1+s (Ω) for some s > 0 even if f C (Ω), under the assumption on the coefficients a ij. Here and thereafter, we use C with or without subscripts to denote a generic positive constant, possibly different at different occurrences, that is independent of the mesh size h and f. By introducing a new variable σ = A u W, we transform the original problem into a system of firstorder (5) with boundary conditions σ +A u = 0 in Ω, σ = f in Ω, (6) u = 0 on Γ D and n σ = 0 on Γ N, Then, the leastsquares method for the firstorder system (5) is: Find u V,σ W such that (7) b(u,σ;v,q) ( σ, q)+ ( A 1 (σ +A u),q+a v ) for all v V,q W. = (f, q),
3 LEASTSQUARES WITH MINIMUM REGULARITY Finite element approximation Let T h be a regular triangulation of Ω (see [7]) with triangular/tetrahedra elements of size h = max{diam(k);k T h }. Let P k (K) be the space of polynomials of degree k on triangle K and denote the local RaviartThomas space of order k on K: RT k (K) = P k (K) n +xp k (K) with x = (x 1,..., x n ). Then the standard (conforming) continuous piecewise polynomials of degree r and the standard H(div) conforming RaviartThomas space of index k [11] are defined, respectively, by (8) (9) V r h = {q V : q K P r (K) K T h }, W k h = {τ W : τ K RT k (K) K T h }. The finite element approximation to (7) is: Find u h V r h and σ h W k h such that (10) b(u h,σ h ;v h,q h ) = (f, q h ), forallv h V r h,q h W k h.notethatthebilinearformb(, ;, )satisfiesthefollowing coercivity property: For any (v,τ) V W, (11) v τ 2 H(div) Cb(v,τ;v,τ). From the above coercivity property and LaxMilgram Theorem, it can be easily shown that (10) has a unique solution. Moreover, the error has the orthogonality property (12) b(u u h,σ σ h ;v h,q h ) = 0, for all v h V r h,q h W k h Mixed and Standard Galerkin methods. We briefly introduce the mixed and standard Galerkin methods. First, define (13) Q r h = {q L2 (Ω) : q K P r (K), for each K T h }. The mixed finite element method corresponding to (5) is defined as follows: Find a pair (u m h,σm h ) Qk h Wk h such that (14) (A 1 σ m h,q h) ( q h,u m h ) = 0, ( σ m h,v h) = (f,v h ) for all (v h,q h ) Q k h Wk h. The pair (u um h,σ σm h ) satisfy the following error equations (15) (16) (A 1 (σ σ m h ),q h ) ( q h,u u m h ) = 0, ( (σ σ m h ),v h) = 0. The standard Galerkin solution u G h V r h is defined as (17) (A (u u G h), ψ) = 0, for all ψ V r h.
4 902 J. KU 4. Upper bound of LS solution with mixed and Galerkin solutions We provide our main result showing that LS methods provide competitive alternatives to mixed and Galerkin methods. In other words, we establish that the LS solution is bounded by the mixed and Galerkin solutions under minimum regularity assumption, i.e. u H 1 (Ω). The proof given here is based on the technique developed in [5]. Theorem 1. Let (u h,σ h ) be the LS solutions satisfying (12), (u m h,σm h ) be the mixed solution satisfying (14), and u G h be the Galerkin solution satisfying (17). Then, (18) σ σ h 0 + u u h 0 C( σ σ m h 0 + u u G h 0 ). Proof. By the triangle inequality and the definition of norms, we have (19) σ σ h 0 + u u h 0 σ σ m h 0 + u u G h 0 + σ m h σ h 0 + u G h u h 0 σ σ m h 0 + u u G h 0 + σ m h σ h H(div) + u G h u h 1. Using coercivity (11) and orthogonal property (12), we have σ m h σ h 2 H(div) + ug h u h 2 1 Cb(u G h u h,σ m h σ h;u G h u h,σ m h σ h) (20) = Cb(u G h u,σ m h σ;u G h u h,σ m h σ h ). Now, by the definition of b( ; ) in (7), and (16), (17), integration by parts and CauchySchwarz inequality, we have b(u G h u,σm h σ;ug h u h,σ m h σ h) = ( (σ m h σ), (σm h σ h)) +(A 1 (σ m h σ +A (ug h u)), σm h σ h +A (u G h u h)) = (A 1 (σ m h σ),σm h σ h)+(σ m h σ, (ug h u h)) (u G h u, (σm h σ h)) C 1 ( σ σ m h u u G h 2 0)+ 1 C 1 ( σ m h σ h 2 H(div) + ug h u h 2 1). Plugging the above inequality into (20) and using a kickback argument for sufficiently large C 1, we obtain σ m h σ h H(div) + u G h u h 1 C( σ σ m h 0 + u u G h 0). Finally, plugging the above estimate into (19), we obtain the desired inequality. This completes the proof. Acknowledgments This paper is motivated by a conversation with Dr. Junping Wang during International Conference on Computational Mathematics, Yonsei University, South Korea. The author would like to thank Dr. Wang for the helpful conversation, and organizers of the conference at Yonsei University for the wonderful opportunities. References [1] P. B. Bochev and M. D. Gunzburger, On leastsquares finite element methods for the Poisson equation and their connection to the Dirichlet and Kelvin principles, SIAM J. Numer. Anal., 43:1 (2005), [2] J. H. Bramble, R. D. Lazarov and J. E. Pasciak, A leastsquares approach based on a discrete minus one inner product for first order systems, Math. Comp., 66(1997), pp [3] J. Brandts, Y. Chen and J. Yang, A note on leastsquares mised finite elements in relation to standard and mised finite elements, IMA J. Numer. Anal., 26 (2006), no. 4, [4] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, SpringVerlag, Berlin, 1991.
5 LEASTSQUARES WITH MINIMUM REGULARITY 903 [5] Z. Cai and J. Ku, Optimal Error Estimate for the Div LeastSquares Method with Data f L 2 and Application to Nonlinear Problems, SIAM J. Numer. Anal., 47 (2010), no. 6, [6] Z. Cai, R. Lazarov, T. A. Manteuffel and S. F. McCormick, Firstorder system least squares for secondorder partial differential equations:part I., SIAM J. Numer. Anal. 31 (1994), pp [7] P. G. Ciarlet, The Finite Element Methods for Elliptic Problems, NorthHolland, Amsterdam, [8] V. Girault and P. A. Raviart, Finite Element Methods for NavierStokes Equations: Theory and Algorithms, SpringerVerlag, New York, [9] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, [10] A. I. Pehlivanov, G. F. Carey and R. D. Lazarov, Leastsquares mixed finite element methods for secondorder elliptic problems, SIAM J. Numer Anal. 31 (1994), pp [11] P.A. Raviart and J. M. Thomas, A mixed finite element method for second order elliptic problems, Mathematical Aspects of the Finite Element Method, Lectures Notes in Math. 606, SpringVerlag, New York(1977). [12] A. Schatz and J. Wang, Some new error estimates for RitzGalerkin methods with minimal regularity assumptions, Math. Comp., 65(1996), pp Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078, USA URL: jku/
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