A Weak Galerkin Method with an Over-Relaxed Stabilization for Low Regularity Elliptic Problems

J Sci Comput (07 7:95 8 DOI 0.007/s095-06-096-4 A Weak Galerkin Metod wit an Over-Relaxed Stabilization for Low Regularity Elliptic Problems Lunji Song, Kaifang Liu San Zao Received: April 06 / Revised: 9 August 06 / Accepted: 6 September 06 / Publised online: 7 September 06 Springer Science+Business Media New York 06 Abstract A new weak Galerkin (WG finite element metod is developed and analyzed for solving second order elliptic problems wit low regularity solutions in te Sobolev space W,p ( wit p (,. A WG stabilizer was introduced by Wang and Ye (Mat Comput 83:0 6, 04 for a simpler variational formulation, and it as been commonly used since ten in te WG literature. In tis work, for te purpose of dealing wit low regularity solutions, we propose to generalize te stabilizer of Wang and Ye by introducing a positive relaxation index to te mes size. e relaxed stabilization gives rise to a considerable flexibility in treating weak continuity along te interior element edges. Wen te norm index p (, ], we strictly derive tat te WG error in energy norm as an optimal convergence order O( l+ p p 4 by taking te relaxed factor β = + p p, and it also as an optimal convergence order O( l+ p in L norm wen te solution u W l+,p wit p [, + p p ] and l. It is recovered for p = tat wit te coice of β =, error estimates in te energy and L norms are optimal for te source term in te sobolev space L. Weak variational forms of te WG metod give rise to desirable flexibility in enforcing boundary conditions and can be easily implemented witout requiring a sufficiently large penalty factor as in te usual discontinuous Galerkin metods. In addition, numerical results illustrate tat te proposed WG metod wit an over-relaxed factor β( converges at optimal algebraic rates for several low regularity elliptic problems. Keywords Weak Galerkin Finite element metods Discrete weak gradient Second order elliptic problems Low regularity B Lunji Song song@lzu.edu.cn Scool of Matematics and Statistics, and Key Laboratory of Applied Matematics and Complex Systems (Gansu Province, Lanzou University, Lanzou 730000, People s Republic of Cina Department of Matematics, University of Alabama, uscaloosa, AL 35487, USA 3

96 J Sci Comput (07 7:95 8 Introduction e goal of tis paper is to formulate and analyze a weak Galerkin metod for solving an elliptic problem wit low regularity solution. We consider a second order elliptic equation wit a omogeneous Diriclet boundary condition as follows: A u = f, in, (. u = 0, on, (. were R is an open bounded polygonal domain, te coefficient A is a positive piecewise constant or a uniformly symmetric positive definite function matrix A := {a ij (x}, a ij = a ji W, ( in R, i.e., tere exists a positive constant γ>0 suc tat a ij (xξ i ξ j γ ( ξ + ξ, ξ := (ξ,ξ R, x. ij= And te force term f L p ( is a given function for some p in (, ]. We sall use standard notation for Sobolev spaces (see []. For any nonnegative integer s and r, te classical Sobolev space on a bounded domain D R is W s,r (D = { v L r (D n v L r (D, n s }, were n v are te partial derivatives of v of order n and L r (D is te space of all (scalarvalued functions on D for wic te corresponding L p -norm { ( /p, v L p (D = D v(x p dx p [,, ess sup x D v(x, p =, is bounded. e corresponding norm in W s,r (D is denoted by W s,r (D and te semi-norm by W s,r (D. e L inner-product is denoted by (, D and by (, if D =. Forte Hilbert space H s (D = W s, (D, te norm is denoted by L (D. We define by H0 (D te subspace of functions in H (D tat vanis on D. rougout te paper, boldface caracters denote vector quantities. e standard weak formulation of (. (. istofindu H0 ( suc tat (A u, v = ( f,v, v H0 (. (.3 Continuous and piecewise polynomials are used in te classical (conforming finite element metods, but te use of discontinuous functions in te finite element approximations often provides muc required flexibility in andling complex problems. ere are a variety of articles wic deal wit discontinuous Galerkin (DG discretizations of elliptic problems under standard regularity assumptions (e.g., for solutions in W, ( or W 3 +ɛ, ( wit ɛ>0 in [,3,3] and te references terein for different types of DG metods. ere are a great deal of difficulties in derivation of algebraic convergence rates for DG metods for low regularity solutions, even toug te numerical flux is treated delicately. Wit solutions belonging to some weigted Sobolev spaces (based on a weigted W, -norm, DG metods ave been analyzed in many papers for elliptic problems wit corner singularities in polygons. In Ref. [6], interior penalty DG metods including symmetric interior penalty Galerkin (SIPG and non-symmetric interior penalty Galerkin (NIPG scemes ave been presented and error analysis as been conducted in an energy norm for solutions in W,p, p (, ]. 3

J Sci Comput (07 7:95 8 97 Recently, wit a well-defined weak gradient, Wang and Ye [5] developed a new weak Galerkin finite element (WG metod for second order elliptic equations formulated as a system of two first order linear equations, and presented optimal a priori error estimates in bot discrete H and L norms, as well as a residual type a posteriori error estimator [4]. In most studies, te weak Galerkin finite element metods are analyzed for analytic solutions defined in te Sobolev space W k+, ( (see [8 0,4]. For ig regularity solutions, a stable and efficient stabilization term as been presented in [5]. Compared to DG finite element metods, WG metods ave some main features, suc as continuous normal flux across element interfaces, less unknowns, and no need for coosing penalty factors. e motivation of te paper is to use a relaxed stabilizer in te weak Galerkin metod to provide an improved approximation to te low regularity solution of problem (. (.. Here we introduce a relaxed power index β on mes size for te stabilization in [5]. Witout adding any penalty factor, particular investigation will be carried out on optimal error estimates of low regularity solutions in te L p space wit < p. ere is a great interest in wat values of te index β would be an optimal coice in numerical analysis for low regularity solutions, and in te question if te coice of β = can be recovered in te case of ig regularity solutions. In tis paper, we will investigate a WG metod wit an over-relaxed stabilization term for solutions existing in H ( W,p ( wit p (, ], and develop a priori error estimates in te standard energy norm and in te L, L p norms. Furtermore, we will prove tat te over-relaxed WG metod converges at an (optimal algebraic rate even if p (, ]. e following Sobolev embedding and regularity results will be used in our analysis. Lemma. ([, eorem 4.] For p [, ], letd R be a bounded open Lipscitz domain. en, te embedding W,p (D L q (D is continuous for all q [,,ifp, and for all q [, p p ] if p <. Lemma. ([7,6] e variational formulation (.3 as a unique solution in W, ( (equivalently, in H ( for any f L p ( wit p (,. Lemma.3 ([6] Given μ>, for any f L p ( wit p (,μ, te solution of (. (. belongs to X := W,p ( H0 ( and te regularity estimate olds u X C f L p (, (.4 were C is a positive constant independent of u. e rest of tis paper is organized as follows. In Sect., we recall te definition of weak gradient and its discrete approximation, and ten present an over-relaxed WG approximation. In Sect. 3, we estimate an a priori WG error in energy norm and some key inequalities are presented. In Sect. 4, error estimates in L p and L norms are derived by a dual argument. In te last Section, we present some numerical experiments including tree low-regularity problems to sow te effectiveness and convergence rates of te over-relaxed weak Galerkin metod. Weak Garlerkin Metods wit an Over-Relaxed Stabilization Let be a triangulation of te domain into any polygons, E be all edges, and E 0 be all interior edges. For any, we denote its diameter by and te boundary by. 3

98 J Sci Comput (07 7:95 8 Furter, let = max denote a caracteristic mes size of te wole partition. We assume tat te partition is sape regular, see [5]. On te partition, we define a broken Sobolev space, for s N and r [, ], by W s,r ( ={v L r ( v W s,r (, }. On an element,wedefineaweakfunctionbyv ={v 0,v b } suc tat v 0 L ( and v b L (, were v 0 and v b represent te values of v in te interior and on te boundary of, respectively. Let P j ( be te set of polynomials on wit degree no more tan j. For a given finite element mes and polynomial degrees j, l, we consider te following finite element spaces V ={v ={v 0,v b }:v P j ( P l (e, e, }, V 0 ={v ={v 0,v b }:v V,v e = 0, e }, U ={v 0 L ( : v 0 P j (, }. Here we take j = l = k for a fixed positive integer k and te function v b is not necessarily te trace of v 0 on. e component v 0 is defined element-wise and may be discontinuous wit respect to v b. e idea of polynomial reduction [] presents an optimal combination for te polynomial spaces. For instance, one can use (P k (, P k (e, [P k ( ] instead of (P k (, P k (e, [P k ( ], to minimize te number of unknowns in te WG sceme witout compromising te accuracy of te stabilized approximations [,]. We denote by Q 0 and Q b te L projection operators from L ( onto P k ( and from L (e onto P k (e, respectively. We write Q ={Q 0, Q b }. Moreover, let Q be te L projection from [L ( ] onto te local discrete gradient space [P k ( ]. e weak gradient is defined by w v [P k ( ] for any function v V satisfying ( w v,q = (v 0, q + v b, q n, q [P k ( ], (. were, stands for te usual inner product in L (. Foranyu ={u 0, u b },v = {v 0,v b } V, we introduce te following bilinear form (A w u, w v := (A w u, w v. Weak Galerkin Algoritm. A numerical approximation for (. (.can be obtained by seeking u ={u 0, u b } V 0 suc tat for any v ={v 0,v b } V 0, a(u,v := (A w u, w v + β u 0 u b,v 0 v b e = ( f,v 0, (. were β is a positive number to be defined later. Next, we justify te well-posedness of te sceme (.. For any v V, we define an energy norm by v := a(v, v. Note tat define a semi-norm in V. However, it defines a norm in V 0.overifytis, it suffices to ceck te positivity property for. o tis end, assume tat v V 0 and v = 0. It follows tat (A w v, w v + β Q b v 0 v b ds = 0, 3 e

J Sci Comput (07 7:95 8 99 wic implies tat w v = 0 on eac element K and Q b v 0 = v b on e. By te definition of te weak gradient, it olds ( w v, τ = ( v 0,τ v 0 v b,τ n e = ( v 0,τ Q b v 0 v b,τ n e. us, v 0 = const on every K. For any edge e E 0, tere exist two elements K and K in suc tat e = K K. is, togeter wit fact tat Q b v 0 K e = v b K e and v b = 0on, implies tat v 0 = v b = 0. Lemma. e weak Galerkin finite element sceme (. as a unique solution. Proof It suffices to prove te uniqueness. If u ( and u ( are two solutions of (., ten e = u ( u ( would satisfy te following equation a(e,v= 0, v V 0. Note tat e V 0. en by letting v = e in te above equation we arrive at e = a(e, e = 0. It follows tat e 0, or equivalently, u ( u (. is completes te proof of te lemma. 3 Error Estimate in Energy Norm Analogously to Lemma 5. in [], te following lemma will be used for multiple times. Lemma 3. Let Q and Q be te L projection operators as defined above. en, on eac element, we ave te following properties: for any v X, τ [P k ( ], (A w (Q v, τ = (Q (A v, τ. (3. Proof It follows from te definition of w, te symmetry of A, and integration by parts tat (A w (Q v, τ = ( w (Q v, Aτ = (Q 0 v, (Aτ + Q b v, (Aτ n = (v, (Aτ + v, (Aτ n = (A v, τ = (Q (A v, τ. Let e ={e 0, e b }={Q 0 u u 0, Q b u u b }.WeaveteerrorequationbetweenQ u and u as follows: a(e,v = (A w e, w v + β e 0 e b,v 0 v b e = ( A u n Q (A u n,v 0 v b e + β Q 0u Q b u,v 0 v b e e E 0 :=l (u,v + l (u,v. (3. 3

00 J Sci Comput (07 7:95 8 Indeed, for any v V 0, testing (. wit v 0, and using integration by parts, te continuity of u n on interior edge e and Lemma 3. leads to ( (A u, v 0 = (A u, v 0 v 0, A u n = (Q (A u, v 0 v 0 v b, A u n + v b, A u n = (Q (A u, w v v 0 v b, A u n Q (A u n e = (A w (Q u, w v v 0 v b, A u n Q (A u n e, (3.3 were in te last second identity we ave used te fact v b = 0and (Q (A u, w v = (v 0, Q (A u + v b, Q (A u n = (Q (A u, v 0 v 0 v b, Q (A u n. Consequently, subtracting (. from (3.3 arrives at (3.. For any two neigboring elements K, K saring an edge e E,let e = ( K K 0 and p (, ]. Note tat if w W,p ( e,tenw is continuous on e; if w W,p ( e,ten w is continuous on e. Lemma 3. For q [, +, and e E,v V, tere olds q (v 0 v b L q (e C (v0 v b L (e, e E,v V, (3.4 equivalently, e v 0 v b q ds C ( e q v 0 v b ds. (3.5 Proof o prove (3.4, we first write ψ = v 0 v b and define a reference element by (ê, ˆψ, ˆPk wit an invertible affine mapping suc tat te two finite elements (e,ψ,p k and (ê, ˆψ, ˆPk affine-equivalent (see te details in [5]. It olds by eorems 3.. and 3..3 in [5] and by norm equivalence for ˆψ q ψ L q (e C ˆψ L q (ê C ˆψ L (ê C ψ L (e. us, (3.4 follows. Next, it suffices to verify (3.5. By using te fact ( n /s ( n /r a r j, for 0 < r s, a j 0, a s j j= j= and taking r =, s = q, we ave ( ( q v 0 v b q ds C v 0 v b ds, e e e E wic completes te proof of (3.5. 3

J Sci Comput (07 7:95 8 0 We need te following lemma of interpolation [6]. Lemma 3.3 Let p (, ], and u W l+,p ( wit l. en tere exists an interpolant : W l+,p ( U suc tat for all tere olds u u W m,p ( C l+ m u W l+,p (, 0 m l +, (3.6 were l is te polynomial degree in te approximation space V. Furtermore, ( ( u u L ( + (u u L C p(l+ ( u p p W l+,p (. (3.7 Now we give te main optimal convergence teorem for te error e in te energy norm. eorem 3. Let u V be te weak Galerkin finite element solution of (. arising from te problem (. (.. Assume tat te exact solution satisfies u W l+,p ( W,p (, were l and p (, ]. en tere exists a constant C suc tat Q u u C ( ( β+ +l p + ( β +l+ p u W l+,p (. (3.8 Wen te index β = + p p is taken, we ave te optimal estimate in te energy norm Especially, if p =, tenβ = and it olds Q u u C l+ p p 4 u W l+,p (. (3.9 Q u u C l u W l+, (. (3.0 Proof In particular, (3.0 follows from (3.8. erefore, it suffices to prove te validity of (3.8. aking v = e in equation (3., we get e = a(e, e = l (u, e + l (u, e. (3. us we need to estimate te upper bounds of te terms l and l.forteterml, by Hölder inequality and te boundedness of A, it olds l (u, e = ( A u Q (A u n, e 0 e b e ( C p c (A u Q (A u n p ( p ds ( ( p C p c ( u Q ( u n p ds β e 0 e b p p ds β e 0 e b p p ds, (3. were p + p = wit p [, and c = p (p β. 3

0 J Sci Comput (07 7:95 8 We ten give te following estimate from Lemma 3. ( β e 0 e b p p ds p c C β p ( C β p ( C ( p (β ( e 0 e b p p ds e 0 e b ds β e 0 e b ds C ( p (β e. (3.3 Applying te trace inequality and scaling argument results in ( u Q u n p L p ( C (p β ( p wic implies by applying (3.6 p c u Q u L p ( + p ( u Q u L p ( p, ( u Q u n p L p ( C en, inserting (3.3 (3.4into(3. results in (p β+pl u p W l+,p (. (3.4 l (u, e C β+ +l p u W l+,p ( e. (3.5 Next, we estimate te bound of te term l. It follows l (u, e = β Q 0u Q b u,v 0 v b e K = C β Q 0u u,v 0 v b e K ( β Q 0u u L (e ( By te trace inequality and elementwise scaling, we get ( β Q 0u u L (e C ( 3 β β v 0 v b L (3.6 (e Q 0 u u L ( + β Q 0 u u L ( C β +(l+ p u W l+,p (, (3.7

J Sci Comput (07 7:95 8 03 were in te last inequality we ave used (3.7. en substituting (3.7 into(3.6gives ( l (u, e C β +(l+ p u W l+,p ( β v 0 v b L (e β +(l+ p u W l+,p ( e. (3.8 Consequently, combining (3. wit (3.5and(3.8 arrives at e C ( β+ +l p + β +(l+ p u W l+,p (. (3.9 is completes te proof of te teorem. Remark 3. Note tat we prove te optimal convergence rate O( l+ p p 4 of te error Q u u in te energy norm, wit β = + p p cosen. Wen p is close to, te optimal rate is up to te order O( l 4 wit β = 5, wic means tat te flexible stabilization term is available for te low regularity solutions. Remark 3. From (3.7, we notice tat te convergence rate β +(l+ p mainly comes from te oscillations of u and u. erefore, wen u is smoot enoug (at least p = and β =, te error estimate in te energy norm maintains optimal, up to O( l,wicis matced wit te teoretical results in [4]. 4 Error Estimate in L p and L Norms In tis section, we mainly derive te error estimate (Q 0 u u 0 in te L p norm and suitable coices for β to be defined later. First, a duality argument will be employed in our analysis for te weak Glerkin finite element approximation. o tis end, we consider a dual problem tat seeks w H 0 ( W,q ( satisfying (A w = Q 0 u u 0 p sign(q 0 u u 0, in, (4. w = 0, on, (4. were p (, ] and Q 0 u u 0 W,p (. Assume tat te above dual problem as te W,q -regularity. Set Y := W,q ( W l+,q ( H0 (. ere exists a constant C suc tat w Y C Q 0 u u 0 p L p (. (4.3 e space H div ( is defined as te set of vector-valued functions on wic, togeter wit teir divergence, are square integrable. We denote by a projection for τ 0 H div ( suc tat τ 0 H div (, and on eac, τ 0 R k ( as well as te following identity ( τ 0,v 0 = ( ( τ 0, v 0, v 0 P k (,, were R k ( is te Raviart-omas element of order k. Lemma 4. For any w W,q ( H 0 ( and u W l+,p (, forwic p + q = and p (, ], tere olds 3

04 J Sci Comput (07 7:95 8 were β [, + p p ]. β ( e b e 0, Q 0 w w e + Q 0 u u, Q 0 w w e C l+ u W l+,p ( w W,q (, (4.4 Proof From te Caucy Scwarz inequality, te definition of, and te trace inequality [5, Lemma A.3], it follows tat β e b e 0, Q 0 w w e ( ( C β e 0 e b L ( β Q 0w w L ( C β ( β C β e ( e 0 e b L ( Q 0w w L ( ( Q 0w w L ( + (Q 0w w L ( ( C ( p +l + ( β+l p u W l+,p ( ( q w W,q ( ( C (l+ + ( β+l p + p u W l+,p ( w W,q (, (4.5 were in te fourt inequality we ave used (3.7andeorem3.. Analogously, wit te use of te Caucy Scwarz inequality and (3.7, we ave β Q 0u u, Q 0 w w e ( C β β Q 0u u ( L ( Q 0w w L ( C β β +(l+ p u W l+,p ( ( q w W,q ( C ( β+l p + p u W l+,p ( w W,q (. (4.6 Due to β [, + p p ] resulting in l++ p p ( β+l p + p (l+, (4.7 q q adding (4.5and(4.6 completes te proof of (4.4. eorem 4. Let u be te weak Galerkin finite element solution to (. arising from te problem (.. Given te factor β [, + p p ], assume te exact solution u W,p ( W l+,p ( wit l, p (, ] and p + q =. In addition, assume tat te dual problem (4. (4. as a W,q ( H0 ( -regularity. en, tere exists a constant 3

J Sci Comput (07 7:95 8 05 C independent of and β suc tat ( Q 0 u u 0 L p ( C f Q 0 f L p ( + l+ u W l+,p (, (4.8 and ( Q 0 u u 0 L ( C p f Q 0 f L p ( + l+ u W l+,p (, (4.9 were C is independent of te mes size and u. Proof esting (4. wit e 0 we obtain e 0 p L p ( = ( (A w, Q 0u u 0 = ( ( A w, Q 0 u u 0 (4.0 It is well known tat v V,τ V ( τ,v 0 = ( τ, w v v b, τ n, wic infers from (4.0 Q 0 u u 0 p L p ( = ( (A w, w (Q u u Q b u u b, (A w n = ( (A w, w (Q u u Q b u, (A w n = ( (A w, w (Q u u u, (A w n = ( (A w, w (Q u u := I 3, (4. were we ave used te fact tat bot (A w and u are continuous across eac interior edge and u b = 0on. anks to te fact (4. follows as I 3 = ( w (Q u, τ = (Q ( u, τ, τ [P k ( ], (4. ( (A w, Q ( u w u = ( (A w A w, Q ( u w u + (A w, Q ( u w u = ( (A w A w, Q ( u w u + (A w, Q ( u w u (4.3 3

06 J Sci Comput (07 7:95 8 Wit te use of te Caucy Scwarz inequality, we estimate te first term of te rigt and side of I 3 in (4.3 ( (A w A w, Q ( u w u ( C (A w A w ( L Q ( ( u w u L. (4.4 ( Applying te Sobolev embedding W,q ( L (D for all, q, and employing a scaling argument yields wic infers And notice tat ( (A w A w L ( C (A w A w L ( 4 q w W,q (, C q w W,q (. Q ( u w u L ( = w (Q u w u L ( C Q u u. en, substituting te estimates above wit eorem 3. into (4.4 and taking β [, + p p ] results in ( (A w A w, Q ( u w u C q Q u u w W,q ( C ( ( β+ +l + ( β +l+ p + p u W l+,p ( w W,q ( C l+ u W l+,p ( Q 0u u 0 p L p (, (4.5 were we ave recalled te following estimates in te last inequality: l++ p p 4 ( β+ +l (l+, l+ p + p ( β +l+ p + p l+ p 4 + p, and (4.5 olds a maximum order wit β+ β +l = +l+ p + p = l++ p p 4 ( l+. 3

J Sci Comput (07 7:95 8 07 For te second term of te rigt and side of I 3 in (4.3, by using (4. and te weak Galerkin formulation (. wit v = Q ( w being taken, we can derive (A w, Q ( u w u = (A w, Q ( u u + (A w, u w u = (A w, Q ( u u + (A w, u (Q (A w, w u = (A w, Q ( u u + (A w, u (A w (Q w, w u = (A w Q (A w, Q ( u u + ( f,w ( f, Q 0 w + β u 0 u b, Q 0 w Q b w e = (A w Q (A w, Q ( u u + ( f Q 0 f,w Q 0 w + β u 0 u b, Q 0 w w e. We bound te terms by te Hölder inequality (A w Q (A w, Q ( u u C A w Q (A w L q ( Q ( u u L p ( C l+ w W,q ( u W l+,p (, (4.6 and ( f Q 0 f,w Q 0 w C w Q 0 w L q ( f Q 0 f L p ( C w W,q ( f Q 0 f L p (. (4.7 It follows from (4.4tat β u 0 u b, Q 0 w w e = β ( (u 0 Q 0 u (u b Q b u, Q 0 w w e + Q 0 u Q b u, Q 0 w w e = β ( e b e 0, Q 0 w w e + Q 0 u u, Q 0 w w e C l+ u W l+,p ( w W,q (. (4.8 3

08 J Sci Comput (07 7:95 8 en combining (4.6 (4.8, we get te following estimate (A w, Q ( u w u ( C C f Q 0 f L p ( + l+ u W l+,p ( ( f Q 0 f L p ( + l+ u W l+,p ( w W,q ( Q 0 u u 0 p L p (. (4.9 Substituting (4.5 and(4.9 into(4.3 yields te desired error estimate (4.8 wit te coice β [, + p p ] from Lemma 4.. Furtermore, (4.9 olds from (4.8 andte fact (see [6] tat for all φ V, φ L ( p φ L p (. is completes te proof of te teorem. 5 Numerical Experiments In tis section, we report on results of numerical tests meant to assess te teoretical a priori error estimates and to illustrate te performance of te over-relaxed WG metod (.wen dealing wit low regularity elliptic problems. In te following numerical studies, all examples will be investigated on uniformly refined triangulations of, and will apply te WG metod to find a solution u = {u 0, u b } were u 0 P (, and u b e P (e. Using te piecewise linear elements (P (, P (e, [P 0 ( ], we test four examples on triangular meses of regular pattern, and te tird example also on locally refined meses. e error for te over-relaxed WG solution of (. sall be measured in te following two norms defined by e = ( e = A w e 0 dx + β e 0 dx. e 0 e b ds We first investigate an example wit a smoot solution to testify te coices of β of te stabilizer in te weak Galerkin metod and apply te incomplete LU (ILU preconditioning to te discrete linear algebraic systems as β>. Example We consider te domain = (0, [ and te elliptic problem (. (. wit x te diffusion coefficient matrix bing given by A = + y ] + xy xy x + y suc tat te + exact solution is u(x, y = sin(π x cos(πy. e errors in te norms e and e as well as te rates of convergence are presented in ables and. Due to te smootness of te solution in W l+, ( wit l =, as β increases, we notice tat te convergence rates in te L and energy norms are optimal wit β =, sowing superconvergence in te energy norm as β =, 3. Figure suggests tat te coice of β greater tan generally results in a better convergence rate in te energy 3,

J Sci Comput (07 7:95 8 09 able Errors for example wit β = 0.5and0.8 β = 0.5 β = 0.8 e e e e /8.7079e+00.464e 0.369e+00 8.8955e 0 /6 9.9850e 0 5.306e 0 7.640e 0.538e 0 /3 5.8586e 0.80e 0 3.858e 0 7.45e 03 /64 3.4495e 0 6.347e 03.044e 0.0696e 03 /8.0365e 0.354e 03.0858e 0 5.96e 04 Rate 0.7669.5083 0.94.808 able Errors for example wit β =,, 3 β = β = β = 3 e e e e e e /8.938e+00 6.43e 0 7.084e 0.5638e 0 4.5774e 0 4.6454e 03 /6 5.9860e 0.6097e 0 3.006e 0.4884e 03.455e 0 4.58e 04 /3.995e 0 4.049e 03.704e 0 3.830e 04 3.83e 0.8648e 04 /64.4978e 0.0063e 03 5.890e 0 5.0844e 05 9.7695e 03 5.530e 05 /8 7.4893e 0.557e 04.033e 0 6.6568e 06.456e 03.440e 05 Rate 0.9988.9996.756.8009.894.9597 0 0 0 0 0 0 0 Error 0 0 H norm (β=0.5 H norm (β=0.8 H norm (β= H norm (β= H norm (β=3 Error 0 3 0 4 0 5 L norm (β=0.5 L norm (β=0.8 L norm (β= L norm (β= L norm (β=3 0 3 0 0 0 0 0 6 0 0 0 0 Fig. Convergence rates for different values of β. Left Error in te H norm. Rigt Error in te L norm norm, but wen β = 3, te WG metod as a comparable convergence rate in te L norm as β =. o attack te ill-conditioned effect from te discrete linear algebraic systems for β>, we employ te ILU preconditioning and a restarted Generalized Minimum Residual metod (GMRES to drive relative residual to less tan a tolerance. All our tests in tis section are started from zero vector and terminated wen te iteration satisfies r (n /r (0 e 6, were r (n is te residual of te n-t iteration. o limit te amount of memory required to execute te metod, we set a restart number up to 00. ables 3 and 4 sow te outer iteration (outer it., inner iteration (inner it. and CPU time of te restarted GMRES metod wit and witout 3

0 J Sci Comput (07 7:95 8 able 3 GMRES metod for example wit β = Non-preconditioning Preconditioning Outer it. Inner it. CPU time (s Outer it. Inner it. CPU time (s /8 7 0.55 43 0.0346 /6 3 4 0.793 7 0.040 /3 8 7 7.7649 5 0.0345 /64 7 9 85.3568 3 0.058 /8 05 73 8988. 4 0.788 able 4 GMRES metod for example wit β = 3 Non-preconditioning Preconditioning Outer it. Inner it. CPU time (s Outer it. Inner it. CPU time (s /8 83 0.985 88 0.0 /6 7 63.588 3 0.0099 /3 9 4 30.606 4 0.030 /64 07 3 36.554 4 0.070 /8 4 49,037.4 0.4688 ILU preconditioning. It is observed tat a preconditioned GMRES metod as produced a very efficient and robust performance. [ ] 0 Example e example is originated from [6]. aking a coefficient matrix A =, 0 we now test te metod for problem (. (. wit te low regularity solution u(x, y = x(x y(y r +α, were α (0, ] is a constant, and r = x + y denotes te distance to te origin. Note tat u W, 0 ( W,p ( for all p (, α (,. As α canges, te errors from te teory in tis work are expected to be Q 0 u u 0 L o( 3 p o( +α, and Q u u o( p p 4 o( 3 α 4 α, were te optimal value of β is + p p. On te uniform triangular meses, we present te errors and convergence rates for different values of β in ables 5, 6, and7, respectively. As α tends to 0, te convergence rates tend to 0 in te energy norm and to in te L -norm for te errors from te WG metod wit β =. Wen β = is taken, te convergence rates in te L and energy norms become optimal for te ig-regularity solution, wic is consistent wit te teory. In Fig., we compare te convergence rates for different β and fixed α (=.It is clear from Fig. tat te overall convergence beavior is very similar to tat of Fig. and te coice β = gives better convergence rates tan te oter two β =, 3. In te case α = 5, it is observed 3

J Sci Comput (07 7:95 8 able 5 Errors for example wit different α =,, 5 and β = 0.5 α =,β = 0.5 α =,β = 0.5 α = 5,β = 0.5 e e e e e e /8.750e 0.4834e 0.7309e+00.4358e 0.43e+0 9.3e 0 /6.738e 0 9.50e 03.470e+00 7.48e 0.3449e+0 6.3854e 0 /3.08e 0 3.465e 03.40e+00 3.635e 0.5733e+0 4.4399e 0 /64 6.736e 0.559e 03.0465e+00.877e 0.8356e+0 3.0788e 0 /8 4.53e 0 4.6040e 04 8.8066e 0 9.086e 03.387e+0.33e 0 Rate 0.68.4387 0.440 0.9958 0.56 0.857 able 6 Errors for example wit different α =,, 5 and β = 0.8 α =,β = 0.8 α =,β = 0.8 α = 5,β = 0.8 e e e e e e /8.979e 0.5737e 0.34e+00 8.6387e 0 8.85e+00 5.4447e 0 /6.659e 0 4.8674e 03.085e+00 3.559e 0 9.3567e+00 3.0948e 0 /3 7.07e 0.484e 03 7.840e 0.4499e 0 9.8665e+00.7484e 0 /64 4.059e 0 4.4648e 04 5.9568e 0 5.8964e 03.0375e+0 9.8486e 0 /8.74e 0.3340e 04 4.579e 0.393e 03.0894e+0 5.5397e 0 Rate 0.890.7 0.397.938 0.0758 0.846 able 7 Errors for example wit different α =,, 5 and β = α =,β = α =,β = α = 5,β = e e e e e e /8.9096e 0.84e 0.39e+00 6.80e 0 7.446e 00 3.8609e 0 /6.0463e 0 3.84e 03 8.300e 0.67e 0 7.350e 00.93e 0 /3 5.6657e 0 8.9899e 04 5.7989e 0 7.9609e 03 7.345e 00 9.403e 0 /64 3.0386e 0.43e 04 4.3e 0.8333e 03 7.004e 00 4.643e 0 /8.667e 0 6.503e 05.99e 0.0057e 03 6.9587e 00.665e 0 Rate 0.8908.8763 0.4889.4857 0.037.09 in Fig. 3 tat te WG metods wit β = 0.5, 0.8 are not convergent in te H norm but converge slowly wit β =. Furtermore, as te value of β increases from to 3 and te values of α decrease, it is observed in ables 7, 8,and9 tat te WG metods wit β =, 3 produce better convergence rates and accuracy in te energy norm for all values of α and, in te case β =, te metod as te best convergence rate in te L norm just for te smoot solution (α =. Especially, wen α = 5 and β, for te low-regularity solution, te WG metod as te first-order optimal convergence rate in te L norm and orders 0.503, 0.6963 in te energy norm for β = andβ = 3, respectively. Due to te condition numbers in te discrete linear algebraic systems from te WG approximation up to O(, O( 3 and O( 4 for β =,, 3, respectively, te ILU preconditioning is indispensable in our 3

J Sci Comput (07 7:95 8 0 0 α= 0 0 α= 0 0 0 0 Error 0 3 Error 0 3 0 4 0 5 0 6 H norm (β=0.5 H norm (β=0.8 H norm (β= H norm (β= H norm (β=3 0 0 0 0 0 4 0 5 0 6 L norm (β=0.5 L norm (β=0.8 L norm (β= L norm (β= L norm (β=3 0 0 0 0 Fig. Convergence rates for different values of β and α =. Left Error in te H norm. Rigt Error in te L norm α= 5 α= 5 0 0 0 0 0 0 0 0 Error 0 0 3 0 4 H norm (β=0.5 H norm (β=0.8 H norm (β= H norm (β= H norm (β=3 H norm (β=.4608 Error 0 0 3 0 4 L norm (β=0.5 L norm (β=0.8 L norm (β= L norm (β= L norm (β=3 L norm (β=.4608 0 0 0 0 0 0 0 0 Fig. 3 Convergence rates for different values of β and α = 5. Left Error in te H norm. Rigt Error in te L norm able 8 Errors for example wit different α =,, 5 and β = α =,β = α =,β = α = 5,β = e e e e e e /8.0584e 0 3.4354e 03 5.03e 0.57e 0 3.507e 00 7.080e 0 /6 4.994e 0 5.8704e 04.687e 0.483e 03.76e 00.890e 0 /3.5e 0 8.0056e 05.4397e 0 4.9634e 04.5708e 00 4.835e 03 /64 9.57e 03.467e 05 7.7e 0.4957e 04.075e 00.5960e 03 /8 3.994e 03 6.863e 06 4.335e 0 6.5476e 05 7.8353e 0 7.487e 04 Rate.87.35 0.8998.90 0.503.6637 computation. able 0 sows te best convergence rates in te energy and L norms wen some critical values of β are cosen for different values of α = 3, 4, 5, respectively. Moreover, since u W,p ( only, linear elements are investigated for computing. In Fig. 4, te profiles of numerical solutions illustrate tat te solutions ave more slope surfaces close to te origin as te values of α become less. Considering te convergence rates in te energy norm, we compare te WG metods (β =,, 3 wit te non-symmetric interior 3

J Sci Comput (07 7:95 8 3 able 9 Errors for example wit α =,, 5 and β = 3 α =,β = 3 α =,β = 3 α = 5,β = 3 e e e e e e /8 6.096e 0 8.078e 04.364e 0 3.59e 03.356e 00.5049e 0 /6.0439e 0 3.34e 04 9.5540e 0.499e 03 6.974e 0 6.0930e 03 /3 6.874e 03.57e 04 4.397e 0 6.3686e 04 3.866e 0 3.457e 03 /64.600e 03 3.7899e 05.4677e 0.4675e 04.4946e 0.7963e 03 /8.398e 03.065e 05.67e 0 9.0743e 05.044e 0 8.9958e 04 Rate.440.5630 0.964.768 0.6963 0.989 able 0 Errors for example wit α = 3, 4, 5 and optimal values of β α = 3,β =.347 α = 4,β =.44 α = 5,β =.4608 e e e e e e /8.679e 00.7545e 0.947e 00 3.096e 0.30e 00 3.303e 0 /6 9.6057e 0 5.869e 03.709e 00 6.7367e 03.963e 00 7.635e 03 /3 5.739e 0.0948e 03 7.0887e 0.6988e 03 7.943e 0 3.0473e 03 /64 3.4689e 0.079e 03 4.3839e 0.4439e 03 4.9453e 0.6684e 03 /8.974e 0 5.3495e 04.8664e 0 7.40e 04 3.86e 0 8.77e 04 Rate 0.730.385 0.6937.987 0.6784.608 penalty Galerkin (NIPG, symmetric interior penalty Galerkin (SIPG and continuous finite element (FEM metods by linear elements presented in Ref. [6], and obtain comparable results in able. It is observed tat wen β =, 3, te WG metods give more impressive convergence rates tan te oter metods. Example 3 e next example is an elliptic problem of corner singularities in te L-saped domain = (0, \[/, wit A = I, an identity matrix. Under a polar coordinate, te solution is system (r,θwit te origin (, u(r = r 3 sin ( θ π 3, π θ π. Note tat te solution in example 3 as a corner singularity at te node (/, / as well as te oter five vertices of te L-saped domain. Wit te reentrant corner of te interior angle 3π/. erefore, te solution as te global regularity H 5 3 ɛ (, wereɛ is any positive number and p =. Some tests are made on uniform grids and locally refined grids to investigate errors and convergence rates. From able, we observe tat as te values of β increase from 0.5 up to, te weak Galerkin metod wit β = as optimal convergence rates in te L and energy norms for te singular problem. e WG solutions for β =, 3 ave better accuracy and convergence rates in te energy norm in able 3, altoug te convergence rate of te WG metod wit β = intel norm is te best from ables and 3. We also employ locally refined grids to illustrate te numerical error in Fig. 5, and verify convergence rates σ of te error e wit respect to te number of degrees of freedom (Dof, defined by 3

4 J Sci Comput (07 7:95 8 Fig. 4 Comparison of numerical solutions wit different values of α =,,, 3, 4, 5, respectively, listed in order from left to rigt and from top to bottom 3

J Sci Comput (07 7:95 8 5 able Comparison on convergence rates of e by using different metods for example wit different α α SIPG in [6] NIPG in[6] FEM in[6] WG (β = WG (β = WG (β = 3 0.905 0.98 0.94 0.8909.87.440 0.49 0.494 0.500 0.4889 0.8998 0.964 0.45 0.47 0.49 0.44 0.7039 0.804 3 0. 0. 0.4 0.75 0.597 0.7439 4 0.0587 0.060 0.068 0.0550 0.536 0.79 able Errors for example 3 wit β = 0.5, 0.8and. β = 0.5 β = 0.8 β = e e e e e e /8.730e 0 7.699e 03.745e 0 6.956e 03.6984e 0 6.75e 03 /6.4e 0.645e 03.097e 0.5437e 03.0954e 0.43e 03 /3 7.978e 0 9.960e 04 7.0e 0 9.55e 04 6.998e 0 8.963e 04 /64 4.5857e 0 3.85e 04 4.53e 0 3.6458e 04 4.4464e 0 3.3794e 04 /8.9097e 0.4866e 04.8764e 0.485e 04.859e 0.976e 04 Rate 0.6437.3975 0.6443.4034 0.6486.450 able 3 Errors for example 3 wit β = and3 β = β = 3 e e e e /8.5094e 0 4.654e 03.0853e 0.3067e 03 /6 8.745e 0.094e 03 4.677e 0.4684e 03 /3 4.848e 0.905e 04.6e 0 7.6808e 04 /64.546e 0.76e 04.804e 0 3.365e 04 /8.33e 0 9.7637e 05 7.3e 03.3897e 04 Rate 0.8830.380 0.9849.033 e := O(Dof σ. (5. able 4 sows te WG metod as better approximation beavior in te locally refined grids tan in te uniform meses, and te coice of β = gives te best convergence rate in te energy norm. Example 4 In tis case, we employ te same analytic solution as in Example and in te domain wit a narrow line crack of size e-5 (see Fig. 6, defined by = (, \[, 0.0000] ( 0.0000, 0.0000. We notice tat te problem wit a Diriclet boundary condition as low regularity and singularity at te corners of te origin. In Fig. 7, it is observed tat te solution around te line crack is discontinuous and as sarp slopes along te bottom-left diagonal direction, but te error mainly distributes around 3

6 J Sci Comput (07 7:95 8 x 0 3 0.8 0 0.6 u u 4 0.4 6 0. 0 0 0. 0.4 0.6 0.8 8 0.8 0.6 0.4 y 0. 0 0 0. 0.4 x 0.6 0.8 Fig. 5 A locally refined grid (left and error profile in 3D (rigt wit β = able 4 Convergence rates of e and e wit respect to Dof for example 3 on locally refined grids, wit β =,, 3 Dof β = β = β = 3 e e e e e e 56 4.9e 0 5.6648e 04.7865e 0.6550e 04.74e 0.586e 04 068.7504e 0.574e 04.943e 0 7.9844e 05 4.8508e 03 8.9586e 05 40880.5699e 0 4.567e 05 5.86e 03 3.030e 05.6496e 03 3.3099e 05 63936 9.79e 03.498e 05.4084e 03.953e 05.684e 03.5e 05 656576 5.4737e 03 4.756e 06.4e 03 4.7085e 06.037e 03 4.876e 06 σ 0.3948 0.8603 0.565 0.764 0.43 0.70 x 0 4.5 0.5 0 0.5.5 5 4 3 0 3 4 5 0 0 8 6 4 0 x 0 4 Fig. 6 An initial grid wit a crack (left and a locally zoomed area around te origin (rigt 3

J Sci Comput (07 7:95 8 7 Fig. 7 Numerical solution in 3D (left and te corresponding numerical error profile (rigt wit te initial grid refined by tree times able 5 Errors for example 4 wit β =,.4608 and 3 on te uniform grids Dof β = β = 3 β =.4608 e e e e e e max{} 5.8e 0.030e 0 5.93e 0 5.855e 0 5.07e 0 7.868e 0 max{}/ 3.63e 0 3.50e 0 3.683e 0.7604e 0 3.60e 0.469e 0 max{}/4.9709e 0.098e 0.964e 0 9.368e 03.9687e 0 9.544e 03 max{}/8.53e 0 5.3365e 03.89e 0 7.0780e 03.475e 0 6.4784e 03 max{}/6.59e 0 3.9304e 03.456e 0 5.5536e 03.504e 0 5.670e 03 Rate 0.440.030 0.440 0.7760 0.457 0.949 te origin. From able 5, it is sown tat wen α = 5, te rate in te L norm wit β =.4608 is better tan tat wit β = 3, and te convergence rates in te energy norm are comparable in te tree cases to te low regularity solutions in te cracked domain. All numerical examples above are in good agreement wit te teoretical analysis, wic validates optimal convergence rates of te stabilized WG finite element metod (. wit te suitable coices of te over-relaxed factor. 6 Conclusions In tis work, we ave proposed and analyzed te a priori energy-norm and L p, L error estimates of te over-relaxed weak Galerkin metod for solving low regularity elliptic problems. In te cases of low regularity elliptic solutions, an over-relaxed factor β> in te over-relaxed stabilization term as been stated wit respect to p (, to implement weak continuity in te WG metod. e WG metod wit te over-relaxed stabilization is optimally convergent, and te rates exibit an impressive performance in te energy norm. e optimal relaxed factor for p (, as been derived and in te case p =, optimal error estimates in te energy and L norms can be recovered wen β = is taken. e relaxed features for low regularity solutions ave been verified by some numerical results. Furtermore, an ILU preconditioning tecnique for te over-relaxed WG sceme is employed troug te restarted GMRES metod to reduce iterations and save computational cost. 3

8 J Sci Comput (07 7:95 8 Acknowledgements e first autor acknowledges support by te Natural Science Foundation of Gansu Province, Cina (Grant 45RJZA046 and Special Program for Applied Researc on Super Computation of te NSFC-Guangdong Joint Fund (te second pase. And te tird autor was supported in part by National Natural Sciences (NSF DMS-38898, and te University of Alabama Researc Stimulation Program (RSP award. References. Adams, R.A.: Sobolev spaces. Academic Press, New York (975. Arnold, D.N.: An interior penalty finite element metod wit discontinuous elements. SIAM J. Numer. Anal. 9, 74 760 (98 3. Castillo, P., Cockburn, B., Perugia, I., Scötzau, D.: An a priori error analysis of te local discontinuous Galerkin metod for elliptic problems. SIAM J. Numer. Anal. 38, 676 706 (000 4. Cen, L., Wang, J., Ye, X.: A posteriori error estimates for weak galerkin finite element metods for second order elliptic problems. J. Sci. Comput. 59(, 496 5 (04 5. Ciarlet, P.G.: e finite element metod for Elliptic Problems. Nort-Holland, Amsterdam (978 6. Crouzeix, M., omee, V.: e stability in L p and W p of te L -projection onto finite element function spaces. Mat. Comput. 48, 5 53 (987 7. Lorenzi, A.: On elliptic equations wit piecewise constant coefficients. II. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Sr. 3 6(4, 839 870 (97 8. Mu, L., Wang, J., Wang, Y., Ye, X.: A computational study of te weak Galerkin metod for second-order elliptic equations. Numer. Algoritms 63, 753 777 (0 9. Mu, L., Wang, J., Wei, G., Ye, X., Zao, S.: Weak Galerkin metods for second order elliptic interface problems. J. Comput. Pys. 50, 06 5 (03 0. Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element metods for te biarmonic equation on polytopal meses. Numer. Metods Partial Differ. Eqs. 30(3, 003 09 (04. Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element metod for second-order elliptic problems on polytopal meses. Int. J. Numer. Anal. Model, 3 53 (05. Mu, L., Wang, J., Ye, X.: A weak Galerkin finite element metod wit polynomial reduction. J. Comput. Appl. Mat. 85, 45 58 (05 3. Rivière, B., Weeler, M.F., Girault, V.: A priori error estimates for finite element metods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39(3,90 93 (00 4. Wang, J., Ye, X.: A weak Galerkin finite element metod for second-order elliptic problems. J. Comput. Appl. Mat. 4, 03 5 (03 5. Wang, J., Ye, X.: A weak Galerkin weak Galerkin mixed finite element metod for second order elliptic problems. Mat. Comput. 83, 0 6 (04 6. Wiler,.P., Rivière, B.: Discontinuous Galerkin metods for second-order elliptic PDE wit lowregularity solutions. J. Sci. Comput. 46, 5 65 (0 3