A priori error estimates of a local-structure-preserving LDG method

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1 A priori error estimates of a local-structure-preserving LDG metod Fengyan Li Department of Matematical Sciences Rensselaer Polytecnic Institute Troy, NY 12180, USA lif@rpi.edu May 22, 2011 Abstract: In tis note, te missing error estimates in te L 2 norm and in a new energy norm are establised for a local-structure-preserving local discontinuous Galerkin metod proposed in [F. Li and C.-W. Su, Metods and Applications of Analysis, v13 (2006), pp ] for te Laplace equation. Wit its distinctive feature in using armonic polynomials as local approximations, te metod as lower computational complexity tan standard discontinuous Galerkin metods. Te analysis in tis note is based on te primal formulation of te sceme. 1 Introduction Te primary goal of tis note is to establis te missing L 2 error estimate for a localstructure-preserving local discontinuous Galerkin (LDG) metod in [6] for solving te Laplace equation. Te metod is based on te standard LDG metod for elliptic equations ([2]), and its distinctive feature of te metod is te use of armonic polynomials (polynomials wic satisfy u = 0) as local approximations for te solution u. Tat is, te numerical solution satisfies te differential equation exactly in eac mes element. Using suc a localstructure-preserving approximation space significantly reduces te size of te matrix in te final system and terefore te overall computational complexity. In [6], error estimates in te energy norm were establised for te local-structure-preserving LDG (LSP-LDG) metod, and tey are confirmed numerically. Numerical experiments also suggest an optimal L 2 error estimate. In tis note, tis missing L 2 error estimate will be proved. Toug one can still work wit te original flux formulation of te bilinear form of te sceme as in [6], te proof in tis note is instead based on te primal formulation ([1]), and tis leads to a more concise analysis for te L 2 error estimate. In particular, te sceme in its primal formulation is bounded and coercive wit respect to a new energy norm B (defined in (3.6)), and it is also consistent and adjoint consistent. Togeter wit te approximation properties of te locally armonic polynomial functions for te solution u of te Laplace equation, te optimal error estimates in bot te L 2 norm and te new energy norm B will come naturally. 1

2 Te analysis in tis note is closely related to te analysis in [1] wic is to study te standard LDG metod in its primal formulation. Te main difference is in te definition of energy norms. Wit te new energy norm B, wic is different from te one in [6] (defined below by (2.12)) and is often used to analyze te (symmetric) interior penalty metod ([5]), one avoids many tecnical complications wic would oterwise be needed. Moreover, te analysis in tis note provides a more straigtforward understanding tan [6] for te role of te approximating space of te auxiliary variable q = u in terms of te accuracy of te LSP-LDG metod. Te rest of te note is organized as follows. In Section 2, bot te standard and LSP-LDG metods are reviewed for solving te Laplace equation. Te optimal error estimates in te L 2 norm as well as in a new energy norm B are establised in Section 3. 2 Review of te metods In tis section, te standard ([2]) and te local-structure-preserving LDG ([6]) metods will be reviewed for te Laplace equation u = 0 in, u ΓD = g D, u n Γ N = g N n, (2.1) were R d is a bounded domain wit n as te unit outer normal along te domain boundary Γ = Γ N Γ D, and Γ D R d 1 > 0. Toug numerical metods presented below can be formulated for general space dimension, tey are discussed and analyzed only for d = 2 in tis note. We start wit a partition T = {K} of te domain, wit te triangular or rectangular element being denoted as K, te edge as e, te diameter of K as K, and te messize of T as = max K T K. We furter denote te union of all interior edges as E i, te union of boundary edges in Γ D (resp. Γ N ) as E D (resp. E N ), and E = E i E D E N. Wit an auxiliary variable q = u, (2.1) can be rewritten as q = u, q = 0 in, u ΓD = g D, q n ΓN = g N n. (2.2) Based on [2], te LDG metod for solving (2.2) can be formulated as: finding (u, q ) (V, M ), suc tat K q rdx = u rdx + û r n K ds, (2.3) K K q vdx = vˆq n K ds (2.4) K 2 K

3 for all (v, r) (V, M ) and K T. Here n K is te unit outer normal of K, (V, M ) is te discrete space pair to approximate (u, q), and (û, ˆq ) are te so-called numerical fluxes, wic are single-valued functions approximating te trace of (u, q) on E. To finalize te sceme, one needs to specify te definitions of (û, ˆq ) and (V, M ). Te LSP-LDG metod in [6] uses te same numerical fluxes defined in [2] for te standard LDG metod, namely, for te interior edge e E i, ˆq = {{q }} C 11 [u ] C 12 [q ], û = {{u }} + C 12 [u ], (2.5) and on te boundary, { q + ˆq = C 11(u + g D)n for e E D, g N for e E N, û = { gd for e E D, u + for e E N. (2.6) Here te standard notations are used for te average {{ }} and te jump [ ]. Suppose e E i and e = K + K, n + (resp. n ) is te unit outer normal of K + (resp. K ), and (v +, r + ) (resp. (v, r )) is te trace of te piecewise smoot function (v, r) from te interior of K + (resp. K ) along e. Ten on tis edge e, {{v}} = (v + + v )/2, {{r}} = (r + + r )/2, (2.7) [v ] = v + n + + v n, [r] = r + n + + r n. (2.8) In addition, for e E D and e K, we define [v ] = v K n, and {{r}} = r K on e wit n being te unit outward normal along Γ. C 11 and C 12 in (2.5)-(2.6) are edgewise defined, and teir values may affect accuracy and stability of LDG metods as well as te matrix structures in te final algebraic system, see [2] and [6]. Detailed assumptions are made in Section 3 for C 11 and C 12. Wit te definition of (û, ˆq ) in (2.5)-(2.6), by summing up all K T, te LDG metod becomes: finding (u, q ) (V, M ), suc tat a(q, r) + b(u, r) =F (r), r M, (2.9) b(v, q ) + c(u, v) =G(v), v V. (2.10) Here a(q, r) = q rdx, c(u, v) = C 11 [u] [v ]ds, E i E D b(u, r) = u rdx ({{u}} + C 12 [u])[r]ds K T K E i = u rdx + K T K F (r) = g D r nds, E D G(v) = ur nds, E N ({{r}} C 12 [r]) [u]ds + ur nds, E i E D C 11 g D vds + vg N nds. E D E N 3

4 Te sceme (2.9)-(2.10) can also be written in a more compact form: finding (u, q ) (V, M ), suc tat A(q, u ; r, v) = F(r, v) (v, r) (V, M ) (2.11) wit A(q, u; r, v) = a(q, r) + b(u, r) b(v, q) + c(u, v) and F(r, v) = F (r) + G(v). Te only difference between te standard ([2]) and te local-structure-preserving ([6]) LDG metods is in te coice of te discrete spaces (V, M ). metod, (V, M ) = (V k,st D, M k,st D ) is used, wit V k,st D = {u L 2 () : u K P k (K), K T }, M k,st D = {q [L 2 ()] d : q K [P k (K)] d, K T }, For te standard LDG were P k (K) is te set of polynomials of degree at most k on K. For te LSP-LDG metod, we take (V, M ) = (V k,lsp, M k,lsp ), wit V k,lsp = {u L 2 () : u K P k (K), u K = 0, K T }, M k,lsp = {q [L 2 ()] d : q K [P k (K)] d, q K = 0, K T }. Note tat q can be solved locally in terms of u, te size of te final algebraic system of te LDG metod terefore depends only on te dimension of V. In [6], anoter discrete space pair, (V, M ) = (V k,lsp k,lsp k,lsp, M ) wit M = M k,st D, is also considered in te LSP-LDG metod. In bot cases, te approximating functions in V for LSP-LDG metods are piecewise armonic polynomials, and suc functions satisfy te differential equations exactly in eac mes element K. Te following is a summary of te results for te standard and te local-structurepreserving LDG metods related to te contents of tis note ([2, 6]). (Solvability) Te LDG metod wit any of te above discrete space pairs is uniquely solvable wen C 11 > 0 on E i E D. Te positiveness of C 11 can be furtered relaxed as in [4] for te standard LDG metod. (Error estimates) Assume te meses {T } are regular (see Section 3). For te sufficiently smoot exact solution (u, q) of (2.2), wit C 12 = O(1), and wit C 11 = O(1) or O( 1 ), te LDG metod equipped wit any of te above discrete space pairs (wit te index k) satisfies te following error estimate (q q, u u ) A = O( k ). 4

5 Here and it defines an energy norm. ) 1/2 (q, u) A = ( q 2L2() + C 11 [u] 2 ds (2.12) E i E D te numerical solutions u V k,st D computed by te standard LDG metod are proved to be optimal in te L 2 norm wen C 11 = O( 1 ) wit u u L 2 () = O( k+1 ), and sub-optimal wen C 11 = O(1) wit u u L 2 () = O( k+1/2 ). Numerical results owever sow optimal convergence rates for bot coices of C 11, see [2, 6]. Suc optimal error estimates are also indicated by te numerical experiments for te LSP-LDG approximation u V k,lsp in te L 2 norm [6]. (Computational complexity) Since te size of te final algebraic system of te LDG metod only depends on te dimension of te space V, by incorporating te a priori knowledge of te exact solution to tis discrete space, te LSP-LDG metod results in a smaller linear system, especially wen iger degree polynomials are used as approximations, and terefore as lower computational complexity. In particular, te dimension of te local-structure-preserving space V k,lsp on eac element K T is 2k + 1 wic depends on k linearly, wereas te dimension of te standard polynomial space V k,st D on K is (k + 2)(k + 1)/2, wic depends on k quadratically. Indeed, tis local-structure-preserving approximating space V k,lsp can be used in any of te DG metods discussed in [1] to provide ig order numerical metods for te Laplace equation wit low computational complexity. Te actual cost efficiency of suc metods certainly needs additional investigation. 3 Error estimates in te L 2 and te energy norms In tis section, te missing optimal L 2 error estimate from [6] will be establised for te LSP-LDG metod wen C 11 = O(1/). Toug one can still work wit te flux formulation of te bilinear form of te sceme (2.9)-(2.10), te proof of tis section is instead based on its primal formulation ([1]), and tis leads to a more concise analysis for te L 2 error estimate. In particular, in te primal formulation, te sceme is bounded and coercive wit respect to a new energy norm B (defined in (3.6)), and it is also consistent and adjoint consistent. Togeter wit te approximation properties ([6]) of te locally armonic polynomial functions for te solution u of te Laplace equation, te optimal error estimates in bot te L 2 norm and te energy norm B will follow. We will comment te L 2 error analysis based on te flux formulation at te end of tis section, 5

6 We first start wit te notations and te assumptions needed in tis section. For C 12 and C 11 in (2.5)-(2.6), it is assumed tat C 12 L (E i ) < η <, C 11 = η e 1 e on e E i E D wit η e as a constant, and min e Ei E D η e > η 0 > 0, max e Ei E D η e < η <. And η 0, η and η are constants independent of. Moreover, te meses {T } are regular, tat is, tere exists a positive constant σ independent of suc tat K ρ K σ, K T (3.1) were ρ K is te diameter of te largest circle B K inscribed in K, and K is star-saped wit respect to B K. If more general quadrilateral elements Ks are used, (3.1) needs to be sligtly revised (p.247 in [3]). In order to get te optimal error estimate in te L 2 norm, te full elliptic regularity is assumed, ψ 2, C r f 0,, f L 2 () (3.2) wit te constant C r solely depending on, for te adjoint problem of (2.1): ψ = f in, ψ ΓD = 0, ψ n Γ N = 0. (3.3) Here H m (D) denotes te Sobolev spaces of order m on D R d, wit te usual norm m,d and H 0 (D) = L 2 (D). We also use V = {v H k+1 () : v = 0 in } and V = V k,lsp. For any v V, v represents its piecewise-defined gradient, and te collection of all suc functions is V. In addition, two assumptions are made to M, te discrete space for te auxiliary variable q = u: (i) Inclusion relation: V M. (3.4) (ii) Inverse inequality: tere exists a constant C > 0, depending on σ (and possibly te index k) and independent of K T and, suc tat Wen V = V k,lsp M k,lsp, and r 0, K C 1/2 K r 0,K, r M. (3.5), examples of M satisfying (3.4) and (3.5) include M k,lsp k 1,LSP M. Different from (2.12) wic was used in [2] and [6], a new energy norm B v 2 B = v 2 0, + v 2, wit v 2 =, M k 1,LSP, e E i E D 1 e [v ] 2 0,e (3.6) is introduced for V () = V + V. Here 0,e is te L 2 norm of a function over an edge e. Note B does not involve te space M. Te analysis in tis section is closely related 6

7 to te one in [1] wic is to study te standard LDG metod in its primal formulation. Te main difference is in te definition of energy norms. Wit v B, we avoid more involved arguments wic oterwise would be needed due to te use of te local-structure-preserving discrete spaces. Trougout te note, te letter C denotes a generic constant (wose value at different occurrences may vary) independent of mes size, but possibly dependent of te sape regularity parameter σ of te mes, te index k (wic is often te polynomial degree) for discrete spaces, te domain, and η 0, η and η related to te numerical fluxes. To carry out te analysis, we first establis several lemmas. Lemma 3.1 (Primal formulation). Te LSP-LDG metod in its flux formulation (2.9)- (2.10) can be rewritten into its primal formulation: Looking for u V suc tat B(u, v) = S(v; g D, g N ), v V, (3.7) were B(w, v) = ( w + t([w ])) ( v + t([v ])) dx + C 11 [w ] [v ]ds, (3.8) E i E D S(v; g D, g N ) = γ(g D ) ( v + t([v ])) dx + C 11 g D vds + vg N nds, (3.9) E D E N = γ(g D ) ( v + t([v ])) dx + G(v). In addition q = u + t([u ]) + γ(g D ). (3.10) Here t([v ]) = α([v ]) + β(c 12 [v ]), and α : [L 2 (E i E D )] d M, β : L 2 (E i ) M, γ : L 2 (E D ) M are lifting operators, defined as follows: r M, α(z) rdx = z {{r}}ds, E i E D β(v) rdx = v [r]ds, E i γ(v) rdx = vr nds. E D Proof. Start wit (2.9), for any r M, one as q rds = u rdx ({{r}} C 12 [r]) [u ]ds u r nds + g D r nds, K T K E i E D E D = u rdx [u ] {{r}}ds + C 12 [u ][r]ds + g D r nds, E i E D E i E D ( = u + α([u ]) + β(c 12 [u ]) + γ(g D ) ) rds. 7

8 Te last inequality results from te definitions of lifting operators. Using te assumption on te inclusion relation (3.4), one obtains te expression of q in terms of u by (3.10). Wit te similar argument, (2.10) turns to ( v + α([v ]) + β(c 12 [v ])) q ds + C 11 [u ] [v ]ds = G(v) (3.11) E i E D for all v V. Now plugging (3.10) into (3.11), one concludes (3.7). Lemma 3.2 (Estimates for lifting operators). For v V (), one as α([v ]) 2 0, C v 2, (3.12) β(c 12 [v ]) 2 0, C C 12 L (E i ) v 2. (3.13) Te constant C depends on σ and k, and it is independent of. Proof. First for any edge e E i E D, one defines a local lifting operator α e : [L 2 (e)] d M, α e (z) rdx = z {{r}}ds, r M. (3.14) e Note tat α e (z) vanises outside te union of one or two elements containing e, and α([v ]) = e E i E D α e ([v ]) for any v V (). By taking r = α e (z) M in (3.14), one as α e (z) 2 0, = z {{α e (z)}}ds z 0,e {{α e (z)}} 0,e, e C 1/2 e z 0,e α e (z) 0,. Te last step is by te inverse inequality assumption (3.5) and by c 0 K e c 1 K for e K, wit te constants c 0, c 1 depending on σ, and C depending on σ and k. Now we ave α e (z) 0, Ce 1/2 z 0,e. And α([v ]) 2 0, = α e ([v ]) 2 0, 3 α e ([v ]) 2 0,, e E i E D e E i E D C 1 e [v ] 2 0,e = C v 2. e E i E D Based on te relation of te lifting operators α( ) and β( ) discussed in [1] (suc relation does not depend on te actual definitions of V and M ), te estimate (3.13) ten follows. Te results in Lemma 3.1 and Lemma 3.2 depend little on te actual definitions of V and M. In fact, Lemma 3.1 only needs te inclusion relation (3.4), and Lemma 3.2 needs te inverse inequality assumption (3.5) on M. 8

9 Lemma 3.3 (Boundedness and coercivity). Te LSP-LDG metod in its primal formulation (3.7)-(3.9) is bounded on V () V () and coercive on V V under te energy norm B for any η 0 = min Ei E D η e > 0. Tat is, (Boundedness) Tere exists a constant M > 0 depending on η, η, σ and k, suc tat B(w, v) M w B v B, w, v V (). (3.15) (Coercivity) Tere exists a constant θ > 0 depending on η 0, η, σ and k, suc tat B(v, v) θ v 2 B, v V. (3.16) Proof. Based on Lemma 3.2, te definition of B(w, v), and te assumptions on C 11 and C 12, te boundedness (3.15) of B(w, v) comes straigtforwardly. To get te coercivity, one starts wit any v V, B(v, v) = v + t([v ]) 2 dx + η e 1 e [v ] 2 0,e e E i E D v + t([v ]) 2 dx + η 0 v 2 (1 ɛ) v 2 0, + (1 1/ɛ) t([v ]) 2 0, + η 0 v 2, ( ɛ (0, 1)) (1 ɛ) v 2 0, + (1 1/ɛ)C 0 v 2 + η 0 v 2, (By Lemma 3.2) = (1 ɛ) v 2 0, + (η 0 + (1 1/ɛ)C 0 ) v 2 Te positive constant C 0 depends on η, σ and k. Now since ɛ can be cosen as close to 1 as one wants, for any η 0 > 0, coose ɛ = ɛ 0 (0, 1) suc tat η 0 + (1 1/ɛ 0 )C 0 η 0 /2, te coercivity result (3.16) will ten follow wit θ = min(1 ɛ 0, η 0 /2). We are now ready to state te main error estimate results. Teorem 3.4 (Error estimate in te energy norm). Let u V be te exact solution and u V = V k,lsp be te numerical solution. Ten we ave u u B C k u k+1,. (3.17) Here te constant C > 0 depends on η 0, η, η, σ and k, and is independent of. As a consequence, for q = u and q by (3.10), we ave q q 0, C k u k+1,. (3.18) 9

10 Proof. Following te derivation of te metod, one can easily see tat te sceme is consistent, terefore te error estimate in te energy norm can be obtained directly based on te boundedness, coercivity of B(w, v) (Lemma 3.3) and te approximation properties of V to V establised in [6]. Tat is, v V, θ v u 2 B B(v u, v u ), (By coercivity) = B(v u, v u ), (By consistency) M u v B v u B, (By boundedness). Ten we ave v u B M θ u v B, v V. Terefore and u u B u v B + v u B (1 + M θ ) u v B u u B (1 + M θ ) inf v V u v B C k u k+1, Te last inequality is from inf v V u v B C 0 k u k+1,, an estimate based on Lemma 3.2, and Corollary 3.3 in [6]. Te constant C 0 depends on σ and k. To get te estimate (3.18) for q q 0,, first notice tat for te exact solution u, one as β(c 12 [u]) = 0 and α([u]) + γ(g D ) = [u] {{r}}ds + E i E D g D r nds E D = un rds + E D g D r nds = 0 E D terefore q = u + α([u]) + β(c 12 [u]) + γ(g D ). Togeter wit (3.10), one gets Tis leads to q q = (u u ) + α([u u ]) + β(c 12 [u u ]). (3.19) q q 0, = (u u ) + α([u u ]) + β(c 12 [u u ]) 0, (u u ) 0, + C 1 u u, (By Lemma 3.2) (C 1 + 1) u u B C k u k+1,, (By (3.17)) C 1 and C are positive constants depending on η 0, η, η, σ and k, and independent of. 10

11 Teorem 3.5 (Error estimate in te L 2 norm). Let u V be te exact solution and u V = V k,lsp be te numerical solution. Ten we ave u u 0, C k+1 u k+1,. Te constant C > 0 depends on η 0, η, η, σ, k and, and is independent of. Proof. Consider te adjoint problem of (2.1). One can easily verify ψ = u u in, ψ ΓD = 0, ψ n Γ N = 0. (3.20) B(v, ψ) = (u u, v) v V (), (3.21) indicating tat te LSP-LDG metod is adjoint consistent. Let ψ I denote te L 2 ortogonal projection of ψ onto V 1,ST D u u 2 0, = B(u u, ψ) = V 1,LSP. By taking v = u u in (3.21), one as = B(u u, ψ ψ I ), (By consistency and ψ I V ) M u u B ψ ψ I B, (By boundedness) MC 0 ψ 2, u u B MC 0 C r u u 0, u u B. Te last two inequalities are based on te approximation property of ψ I to ψ ([3]) and te full regularity assumption (3.2) for (3.20), wit te constant C 0 depending on σ and C r on. Togeter wit Teorem 3.4, one gets te error estimate in te L 2 norm u u 0, MC 0 C r u u B C k+1 u k+1,. Note te analysis in tis section is based on te primal formulation of te sceme, it only relies on te approximation properties of V, not tose of M, terefore te error estimates in bot te energy ( B ) and te L 2 norms from Teorems are optimal wit respect to te approximation properties of V. For te L 2 error estimate, te key observation is V 1,LSP = V 1,ST D, tat is, V 1,LSP can be used to approximate te solutions of bot te Laplace equation and its adjoint problem. As for M, only te assumptions (3.4)-(3.5) are needed. Te optimal L 2 error estimate in Teorem 3.5 indeed can also be establised based on te flux formulation of te sceme wit duality argument ([2]), and te analysis will be more 11

12 involved tan te one presented above. To carry out suc analysis, besides te standard components, one would also need V 1,LSP = V 1,ST D, M 0,LSP = M 0,ST D, and (u u ) 0, = q q 0, + C k u k+1,. Te latter is suggested by Teorem 3.5, and it can be derived from (3.19), Lemma 3.2, and Corollary 3.3 in [6]. Acknowledgment: Te researc was supported in part by te National Science Foundation under te grant DMS , CAREER award DMS and by an Alfred P. Sloan Researc Fellowsip. References [1] D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin metods for elliptic problems, SIAM Journal on Numerical Analysis, v39 (2001/02), pp [2] P. Castillo, B. Cockburn, I. Perugia and D. Scötzau, An a priori error analysis of te local discontinuous Galerkin metod for elliptic problems, SIAM Journal on Numerical Analysis, v38 (2000), pp [3] P. Ciarlet, Te finite element metods for elliptic problems, Nort-Holland, Amsterdam, [4] B. Cockburn and B. Dong, An analysis of te minimal dissipation local discontinuous Galerkin metod for convection diffusion problems, Journal of Scientific Computing, v32 (2007), pp [5] J. Douglas, Jr. and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin metods, Lecture Notes in Pysics, Springer-Verlag, Berlin, v58 (1976), pp [6] F. Li and C.-W. Su, A local-structure-preserving local discontinuous Galerkin metod for te Laplace equation, Metods and Applications of Analysis, v13 (2006), pp