1. Introduction. We consider the model problem: seeking an unknown function u satisfying

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1 A DISCONTINUOUS LEAST-SQUARES FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS XIU YE AND SHANGYOU ZHANG Abstract In tis paper, a discontinuous least-squares (DLS) finite element metod is introduced Te novelty of tis work is twofold, to develop a DLS formulation tat works for general polytopal meses and to provide rigorous error analysis for it Tis new metod provides accurate approximations for bot te primal and te flux variables We obtain optimal order error estimates for bot te primal and te flux variables Numerical examples are tested for polynomials up to degree 4 on non-triangular meses, ie, on rectangular and exagonal meses Key words discontinuous Galerkin, finite element metods, least-squares finite element metods, second-order elliptic problems, polygonal mes AMS subject classifications Primary, 65N15, 65N30, 76D07; Secondary, 35B45, 35J50 1 Introduction We consider te model problem: seeking an unknown function u satisfying (11) (12) (a u) + cu = f, in Ω, u = 0, on Ω, were c 0 and Ω is a polytopal domain in R d wit d = 2, 3, u denotes te gradient of te function u, and a is a d d tensor tat is uniformly symmetric positive-definite in te domain Te partial differential equation (11) is a bencmark testing problem for new discretization tecniques Te goal of tis work is to develop a discontinuous least-squares finite element metod for te second order elliptic problem (11)-(12) and provide rigorous error analysis for te metod Tis new DLS finite element metod as two unique features: 1 provide accurate approximation for bot te primal and te flux variables and lead to a positive and definite system, 2 allow te use of general polytopal meses Te least-squares finite element metod is a discretization tecnique for solving partial differential equations Te metod receives its name by minimizing te residuals in a least-squares fasion Least-squares finite element metods ave been developed for te second-order elliptic problems in [3, 6, 8, 9, 13, 17] and references terein Te researc of finite element metods wit discontinuous approximations as received extensive attention in te past two decades Tousands of papers ave been publised on discontinuous Galerkin tecniques; a few representatives include interior penalty discontinuous Galerkin metod [1], local discontinuous Galerkin metod [12], ybridizable discontinuous Galerkin metod [11] Discontinuous least-squares metods [14, 15] ave been developed for singularly perturbed reaction-diffusion problems Tese metods are te first order metods Department of Matematics, University of Arkansas at Little Rock, Little Rock, AR (xxye@ualredu) Tis researc was supported in part by National Science Foundation Grant DMS Department of Matematical Sciences, University of Delaware, Newark, DE (szang@udeledu) 1

2 2 wit nonsymmetric system Discontinuous least-squares metods ave been developed in [4, 5] for te Stokes equations Optimal or near optimal convergence rates of te metods are only confirmed numerically In [2], a discontinuous least-squares metod is introduced for te div-curl system on tetraedral mes Most of existing least squares finite element metods can only be applied on simplicial mes suc as triangular or quadrilateral mes Recently, a weak Galerkin least squares finite element metod as been developed in [16] for te second order elliptic problems, wic can work on general polytopal mes Weak Galerkin metods refer to general finite element tecniques for partial differential equations involving novel concepts of weak function and weak derivative introduced first in [19, 18] for second order elliptic equations Motivated by te work in [16], we extend te results of te weak Galerkin metod to te discontinuous Galerkin metod in tis paper We develop and analyze a discontinuous Galerkin least squares metod on polygonal/polyedral mes To te best of our knowledge, tis new metod is te first discontinuous Galerkin ig order polygonal least-squares finite element metod for te second order elliptic equations wit rigorous error estimates 2 Discontinuous Galerkin Least Squares Metod Let T be a partition of te domain Ω into polygons in 2D or polyedra in 3D Assume tat T is sape regular in te sense as defined in [18] Denote by E te set of all edges or flat faces in T, and let E 0 = E \ Ω be te set of all interior edges or flat faces Let Γ be te subset of E of all edges or faces on Γ = Ω For every element T T, we denote by T its diameter and mes size = max T T T for T A mixed form of te problem (11)-(12) can be stated as: Find q = q(x) and u = u(x) satisfying (21) (22) (23) q + a u = 0, in Ω, q + cu = f, in Ω, u = 0, on Ω We now introduce two finite element spaces: V for te pressure variable u and Σ for te flux variable q defined as follows (24) { V = {v L 2 (Ω) : v T P k (T ), T T }, Σ = {σ [L 2 (Ω)] d : σ T [P k (T )] d, T T }, were k 1 is any nonnegative integer Let elements T 1 and T 2 ave e as a common edge wit n 1 and n 2 as te unit outward normal respectively We define te jump and te average for a scalar function v on e as { v T1 n [v] e = 1 + v T2 n 2, e E 0, vn, e Γ, { 1 {v} e = 2 (v T 1 + v T2 ), e E 0, v, e Γ

3 3 We define te jump and te average for a vector function σ on e as { σ T1 n [σ] e = 1 + σ T2 n 2, e E 0, σ n, e Γ, { 1 {σ} e = 2 (σ T 1 + σ T2 ), e E 0, σ, e Γ First, we introduce some notations, (v, w) T = (v, w) T = vwdx, T T T T T v, w T = v, w T = vwds, T T T T T v, w E = v, w e = vwds e E e E Ten we define a bilinear form, e (25) A(τ, w; σ, v) =( τ + cw, σ + cv) T + (τ + a w, σ + a v) T + s 1 (w, v) + s 2 (τ, σ), were s 1 (w, v) = 1 e [w], [v] E, s 2 (τ, σ) = 1 e [τ ], [σ] E 0 Algoritm 1 Te discontinuous Galerkin lease-squares metod for te problem (21)-(23) seeks u V and q Σ satisfying (26) A(q, u ; σ, v) = (f, σ + cv), σ v Σ V 3 Existence and Uniqueness Let T be an element wit e as an edge For any function φ H 1 (T ), te following trace inequality olds true, (31) φ 2 e C ( 1 T φ 2 T + T φ 2 T ) Define Introduce a norm V v 2 1, = T T v 2 T for V and a norm Σ for Σ as follows v 2 V = v 2 1, + s 1 (v, v), σ 2 Σ = T T σ 2 T + σ 2 + s 2 (σ, σ) Te following discrete Poincaré inequality as been establised in [7], (32) v C v V, v V

4 4 Lemma 31 Tere exists a constant C suc tat for all σ v Σ V one as (33) A(σ, v; σ, v) C( σ 2 Σ + v 2 V ) Proof First, note tat (34) (35) s 1 (v, v) A(σ, v; σ, v), s 2 (σ, σ) A(σ, v; σ, v) It follows from te integration by parts, (36) ( v, σ) T =(v, σ) T v, σ n T =( σ + cv, v) T (cv, v) T v, σ n T Using te trace and te inverse inequalities, we ave tat for σ v Σ V, (37) v, σ n T = [v], {σ} E + {v}, [σ] E 0 C(s 1/2 1 (v, v) σ + s 1/2 2 (σσ) v ) Te equation (36) implies 2 v 2 T = (a v + σ, v) T (σ, v) T T T a 1 = (a v + σ, v) T + ( σ + cv, v) T (cv, v) T v, σ n T (a v + σ, v) T + ( σ + cv, v) T v, σ n T, and a 1 2 σ 2 = (σ + a v, a 1 σ) T ( v, σ) T = (σ + a v, a 1 σ) T + ( σ + cv, v) T (cv, v) T v, σ n T (σ + a v, a 1 σ) T + ( σ + cv, v) T v, σ n T Using (32) and (37), we ave v 2 V + σ 2 C( T T a 1 2 v 2 T + a 1 2 σ 2 ) + s 1 (v, v) C((σ + a v, v) T + (σ + a v, a 1 σ) T + 2( σ + cv, v) T 2 v, σ n E ) + s 1 (v, v) CA 1/2 (σ, v; σ, v)( v V + σ ) + A(σ, v; σ, v) CA(σ, v; σ, v) ( v V + σ )2 + A(σ, v; σ, v) CA(σ, v; σ, v) ( v 2 V + σ 2 ),

5 5 wic implies (38) v 2 V + σ 2 CA(σ, v; σ, v) Similarly, we ave from te estimates (32) and (38) tat σ 2 C( σ + cv 2 + v 2 ) CA(σ, v; σ, v) Combining te two estimates above wit (35) gives σ 2 Σ CA(σ, v; σ, v), wic, togeter wit (38), completes te proof of te lemma Lemma 32 Te discontinuous least-squares finite element sceme (26) as one and only one solution Proof It suffices to prove te uniqueness If q (1) solutions of (26), ten τ = u (1) u(2) and η = q (1) equation u(1) and q (2) u(2) are two would satisfy te following q(2) A(η, τ ; σ, v) = 0, σ v Σ V Note tat τ V By letting v = τ and σ = η in te above equation we ave τ 2 V + η 2 Σ Ca s(η, τ ; η, τ ) = 0, wic implies τ 0 and η 0 or equivalently, u (1) u (2) and q (1) q (2) Tis completes te proof of te lemma 4 Error Analysis Let q u Σ V be te discontinuous least-squares finite element solution arising from (26), and Q q Q u Σ V be te L 2 projection of te exact solution q u defined element-wise Teir differences are referred to as te error functions, and tey are denoted as (41) ε = Q q q, e = Q u u Lemma 41 Assume tat T is sape regular Ten for u H k+1 (Ω) and q [H k+1 (Ω)] d, we ave (42) (43) s 1 (Q u, v) C k u k+1 v V, s 2 (Q q, σ) C k q k+1 σ Σ, Proof First, recall tat Q and Q are te L 2 projections defined elementwise onto P k (T ) and [P k (T )] d respectively for eac element T T We will use approximation properties of standard L 2 projections in te following estimates Using te trace inequality (31), we obtain s 1 (Q u, v) = 1 e [Q u], [v] E = 1 e [Q u u], [v] E ( C ( 2 Q u u 2 T + (Q u u) 2 T ) T T C k u k+1 v V ) 1 2 v V

6 6 Similarly, we ave s 2 (Q q, σ) = 1 e [Q q q], [σ] E 0 C k q k+1 σ Σ Teorem 42 Let q u Σ V be te discontinuous least-squares finite element solution of te problem (21)-(23) arising from (26) Assume te exact solution u H k+1 (Ω) and q [H k+1 (Ω)] d, ten (44) u Q u V + q Q q Σ C k ( u k+1 + q k+1 ) Proof It is obvious tat te solution (u, q) satisfies A(q, u; σ, v) = (f, σ + cv), σ v Σ V Te difference of te above equation and te equation (26) gives A(q q, u u ; σ, v) = 0, σ v Σ V It follows from te equation above tat A(ε, e ; σ, v) = A(q Q q, u Q u; σ, v) σ v Σ V Letting v = e and σ = ε in te equation above, we ave (45) A(ε, e ; ε, e ) = A(q Q q, u Q u; ε, e ) It ten follows from (33), (42), (43) and te definitions of Q and Q tat e 2 V + ε 2 Σ CA(ε, e ; ε, e ) = C A(q Q q, u Q u; ε, e ) C( ( (q Q q) + c(u Q u), ε + ce ) T + (q Q q + a (u Q u), ε + a e ) T + s 1 (Q u, e ) + s 2 (Q q, ε ) ) C k ( u k+1 + q k+1 )( e V + ε Σ ), wic implies (44) Tis completes te proof Remark 43 In tis paper, bot pressure and velocity variables are approximated by kt order polynomials for k 1 For tis case, it is difficult to prove an optimal L 2 convergence rate of te pressure variable for te least-squares finite element metod We cannot find similar results in literature In [17], an optimal L 2 convergence rate for te pressure is proved wen one degree iger polynomials are used for te flux variable However, te numerical results in te next section demonstrate our metod produces an approximation for pressure u wit an optimal L 2 convergence rate

7 7 5 Numerical Test We numerically solve te boundary value problem (11)- (12) on te unit square Ω = (0, 1) (0, 1) wit te exact solution (51) u = 2 4 x(1 x)y(1 y), were a = 1 and c = 1 So te exact gradient solution is ( ) 2 4 (1 2x)y(1 y) (52) q = 2 4 x(1 x)(1 2y) Fig 51 Te first four grids for numerical solutions in Tables Table 51 Te errors, e u = Q u u and e q = Q q q, and te order of convergence, by te P 1 DLS metod (24) on square grids (Figure 51), for (51) e u 0 n e u n e q 0 n e q n Table 52 Te errors, e u = Q u u and e q = Q q q, and te order of convergence, by te P 2 DLS metod (24) on square grids (Figure 51), for (51) e u 0 n e u n e q 0 n e q n In te first set of tests, we use te DLS metod (24) wit polynomial degree k = 1, 2, 3, 4 on te uniform grids sown in Figure 51 We note tat te polynomial space of separated degree k, Q k, is used in typical finite element metods, but ere we

8 8 Table 53 Te errors, e u = Q u u and e q = Q q q, and te order of convergence, by te P 3 DLS metod (24) on square grids (Figure 51), for (51) e u 0 n e u n e q 0 n e q n Table 54 Te errors and te order of convergence, by te P 4 DLS metod (24) on square grids (Figure 51), for (51) Q u u 0 n Q u u 1 n Q q q 0 n used P k polynomials, taking an advantage of te discontinuous Galerkin metod Te error and te order of convergence are displayed in Tables 51-54, were Q and Q q are te L 2 projections, and u and q te numerical solutions Te optimal order of convergence is acieved in all cases In particular, wen k = 4, te exact solution is obtained, up to te computer accuracy To see conformity, ie, discontinuity, we plot te finite element solutions on te level 4 square grid in Figure 52 Table 55 Te errors, e u = Q u u and e q = Q q q, and te order of convergence, by te P 1 DLS metod (24) on exagon grids (Figure 53), for (51) e u 0 n e u n e q 0 n e q n In te second set of tests, we use te DLS metod (24) wit polynomial degree k = 1, 2, 3, 4 again on te exagon grids sown in Figure 53 Te error and te order of convergence are listed in Tables 55-58, were Q u and Q q are te L 2 projections, and u and q te numerical solutions Te optimal order of convergence is also acieved in all cases Wen k = 4, te exact solution is inside te discontinuous finite element space, and te numerical solution is exact up to te computer accuracy We

9 9 ( 00, 00, ) ( 00, 00, ) ( 10, 10, ) ( 00, 00, ) ( 10, 10, ) ( 10, 10, ) Fig 52 Te finite element solutions, u, (q ) 1 and (q ) 2 of P 1 DLS (24) for (51) on level 4 square grid of Figure 51 Fig 53 Te first four exagon grids for numerical solutions in Tables plot te finite element solutions on te level 4 exagon grids in Figure 54 REFERENCES [1] D Arnold, F Brezzi, B Cockburn and D Marini, Unified analysis of discontinuous Galerkin metods for elliptic problems, SIAM J Numer Anal, 39 (2002), [2] R Bensow and M Larson, Discontinuous Least-Squares finite element metod for te Div-Curl problem, Numer Mat, 101 (2005), [3] P Bocev and M Gunzburger, Least-Squares Finite Element Metods, Springer, New York,

10 10 Table 56 Te errors, e u = Q u u and e q = Q q q, and te order of convergence, by te P 2 DLS metod (24) on exagon grids (Figure 53), for (51) e u 0 n e u n e q 0 n e q n Table 57 Te errors, e u = Q u u and e q = Q q q, and te order of convergence, by te P 3 DLS metod (24) on exagon grids (Figure 53), for (51) e u 0 n e u n e q 0 n e q n Table 58 Te errors and te order of convergence, by te P 4 DLS metod (24) on exagon grids (Figure 53), for (51) Q u u 0 n Q u u 1 n Q q q 0 n [4] P Bocev, J Lai and L Olson, A locally conservative, discontinuous least-squares finite element metod for te Stokes equations, Int J Numer Met Fluids, 68 (2011), [5] P Bocev, J Lai and L Olson, A non-conforming least-squares finite element metod for te velocity-vorticity-pressure Stokes equations, Int J Numer Met Fluids, 00 (2011), 1-19 [6] J Bramble, R Lazarov and J Pasciak, A least-squares approac based on a discrete minus one inner product for first order systems, Mat Comp, 66 (1997), [7] S Brenner, Korns inequalities for piecewise H 1 vector elds, Mat Comput, 73 (2004), [8] Z Cai, R D Lazarov, T Manteuffel and S McCormick, First-order system least squares for partial differential equations: Part I, SIAM J Numer Anal, 31 (1994), [9] Z Cai, T Manteuffel, and S McCormick, First-order system least squares for partial differential equations: Part II, SIAM J Numer Anal, 34 (1997), [10] C L Cang, An error estimate of te least squares finite element metods for te Stokes problem in tree dimensions, Mat Comp, 63 (1994), [11] B Cockburn, J Gopalakrisnan, and R Lazarov, Unified ybridization of discontinuous Galerkin, mixed, and conforming Galerkin metods for second order elliptic problems, SIAM J Numer Anal, 47, (2009),

11 11 ( 00, 00, ) ( 00, 00, ) ( 10, 10, ) ( 00, 00, ) ( 10, 10, ) ( 10, 10, ) Fig 54 Te finite element solutions, u, (q ) 1 and (q ) 2 of P 1 DLS (24) for (51) on te level 4 exagon grid of Figure 53 [12] B Cockburn and C Su, Te local discontinuous Galerkin metod for time-dependent convection-diffusion systems, SIAM J Numer Anal, 35 (1998), [13] B Jiang and L Povinelli, Optimal least-squares finite element metod for elliptic problems, Comput Metods Appl Mec Engrg, 102 (1993), [14] R Lin, Discontinuous discretization for least-squares formulation of singularly perturbed reactive-diffusion problems in one and two dimensions, SIAM J Numer Anal, 47 (2008), [15] R Lin, Discontinuous Galerkin least-squares finite element metods for singularly perturbed reaction-diffusion problems wit discontinuous coefficients and boundary singularities, Numer Mat, 112 (2009), [16] L Mu, J Wang, and X Ye, A least-squares based weak Galerkin finite element metod for second order elliptic equations, SIAM J Sci Comp, 39 (2017), A1531-A1557 [17] A Pelivanov, G Carey and R Lazarov, Least-squares mixed finite elements for second-order elliptic problems, SIAM J Numer Anal, 31 (1994), [18] J Wang and X Ye, A Weak Galerkin mixed finite element metod for second-order elliptic problems, Mat Comp, 83 (2014), [19] J Wang and X Ye, A weak Galerkin finite element metod for second-order elliptic problems, J Comp and Appl Mat, 241 (2013),