WEIGHTED ERROR ESTIMATES OF THE CONTINUOUS INTERIOR PENALTY METHOD FOR SINGULARLY PERTURBED PROBLEMS

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1 WEIGHTED ERROR ESTIMATES OF THE CONTINUOUS INTERIOR PENALTY METHOD FOR SINGULARLY PERTURBED PROBLEMS ERIK BURMAN, JOHNNY GUZMÁN, AND DMITRIY LEYKEKHMAN Abstract. In tis paper we analyze local properties of te Continuous Interior Penalty CIP Metod for a model convection-doated singularly perturbed convection-diffusion problem. We sow weigted a priori error estimates, were te weigt function exponentially decays outside te subdomain of interest. Tis result sows tat locally, te CIP metod is comparable to te Streamline Diffusion SD or te Discontinuous Galerkin DG metods. 1. Introduction Te Continuous Interior Penalty CIP metod was originally proposed by Douglas and Dupont [9] for parabolic and elliptic equations. Te idea was to add a penalization term on te gradient jumps in order to increase robustness for elliptic problems wit a doating convection term. Te case of optimal convergence in te ig Péclet number regime was analyzed by Burman and Hansbo [2] and Burman [3] for first order conforg and non-conforg approximation and in te framework of p-finite elements by Burman and Ern [5]. In tis paper, we are interested in approximating te solution u of te following model problem 1.1 ε u + u x + u = f in, u = 0 on, were is a polygonal domain, 0 < ε 1, and f L 2. Let U denote te approximate solution and te quality of te mes. Typically te error is sown to satisfy u U L 2 C k+ 1 2 u H k+1 in te ig Péclet number regime, were k is te polynomial order and assug u as sufficient regularity. Optimal convergence in of te error in te streamline derivative can also be derived. Tese results are similar to te typical estimates for oter stabilized metods suc as Discontinuous Galerkin DG metod or te Streamline-Diffusion SD metod [13]. Te rigt-and side of tese estimates, owever, depends on a global Sobolev norm. Tis norm may be large in te presence of layers. Terefore te estimates can be considered to be of practical interest only in case te solution is smoot. Date: Feb 07, Matematics Subject Classification. 65N30,65N15. Key words and prases. singularly perturbed, convection-diffusion, local error analysis, continuous interior penalty metod. 1

2 2 BURMAN, GUZMÁN, AND LEYKEKHMAN However, many problems of interest do not ave smoot solutions. Very often tey exibit nonsmoot beavior, like socks, boundary or interior layers and interface discontinuities. Model problem 1.1 for example is known to exibit internal parabolic layer of order O ε log 1/ε and exponential outflow layers of order Oε log 1/ε. In designing a numerical metod it is important to know ow te metod will beave in te neigborood of suc a discontinuity, and weter or not te resulting effects are global or local. For example wen under-resolved layers are present it is well known tat te standard Galerkin metod suffers from oscillations tat pollute te wole solution. One approac to assess te propagation of perturbations for a given metod is to prove tat te local error away from te possibly under-resolved layer as optimal convergence. An appealing way to prove suc results is by using weigted a priori error estimates to prove local error estimates as was done for te Streamline- Diffusion metod by Jonson, Nävert and Pitkäranta in te case of local L 2 -norm estimates [13]. Te analysis was ten extended by Jonson, Walbin and Scatz to L -norm estimates [14]. Recently, similar results were proved in te case of te residual free bubble metod by Sangalli [16] and te Discontinuous Galerkin metod by Guzmán [12]. Te CIP metod is one instance of a class of symmetric stabilization metods tat as received increasing interest lately. Examples of oter members of tis group are te subgrid viscosity metod proposed by Guermond [11], te ortogonal subscale stabilization proposed by Codina and Blasco [8], and te local projection stabilization proposed by Becker and Braack [1]. In tis work we prove weigted a priori error estimates for te CIP metod. To our knowledge tis is te first time suc estimates ave been proved for a metod from te class of symmetric stabilizations. We sow tat te CIP metod as te same upwind and crosswind error propagation properties as te SD-metod. In particular te penalization of te jumps of te crosswind derivative allows an improved estimate of te error in te crosswind derivative in te case of piecewise linear approximation. Compared to te DG-metod or te SD-metod te proof of local estimates for te CIP-metod is more involved. Tis is due to te fact tat te stabilization only controls te part of te streamline derivative tat is not in te finite element space. Terefore te desired control of te streamline derivative is obtained in a more implicit fasion tan for te oter metods. In order to avoid unnecessary tecnicalities we consider te simple model problem 1.1 wit constant convection velocity and reaction terms. Also, for te sake of simplicity, we only give te detailed proof for te case of first order polynomial approximation. However te present analysis extends to te more general case of ig polynomial order and also allows to prove L -norm error estimates. We will comment on tis in te final section of te paper. In te remaining part of tis section we will introduce te CIP-metod for problem 1.1 and we state te main results Te CIP metod. Let {T } be a one-parameter family of face-to-face triangulations of, wit = sup T T T, were T = diamt. Te triangulations are assumed to be globally quasi-uniform, i.e. if necessary after a renormalization

3 WEIGHTED ESTIMATES OF CIP METHOD 3 of, diam T Cmeas T 1/2 T T. Troug out tis paper we assume 0 < < 1. By V we will denote a finite dimensional space of continuous piecewise linear polynomial functions. We will not pose any boundary conditions on V. We let E be te collection of boundary edges, E 0 be te collection of interior edges corresponding to T, and E = E 0 E. Te Continuous Interior Penalty approximation U V is defined as te unique solution to 1.2 BU, v = fv, v V, were Bu, v = εau, v + Mu, v + J u, v + J u, v, wit Au, v = Mu, v = u v u n v + v n u ds + γ bc u x v + uv + uv n x ds, [u x ][v x ] ds, J u, v = 2 e E 0 e J u, v = 2 [u y ][v y ] ds + 1/2 e E 0 e uv n y 2 ds. uv ds, Here n = n x, n y is te outward unit vector to. Te inflow part of te boundary is given by = {x, y : n x x, y < 0}. Te constant γ bc is a boundary penalty parameter tat as to be cosen large enoug in order to guarantee stability. Te coice of 2 in te definition of te stability terms Ju, v is not essential and can be replaced wit oter expressions cf. [3]. Te jump operator is given by [v]x, y = lim t 0 +vx, y tn e vx, y + tn e, were e is a mes interior edge, x, y e, and n e is a fixed unit vector normal to te edge e. Te last term of J does not appear in te original definition of te CIP metod appearing in [3]. We added tis term in order to improve te widt of te crosswind layer to order maxε 1/2, 3/4 log1/. Witout tis term we can only prove tat te widt of te crosswind layer is of order 1/2 log1/. By te regularity teory u H 3 2 +δ for some δ > 0 cf. [10]. Hence u satisfies 1.2 and we ave te usual Galerkin ortogonality property 1.3 Bu U, v = 0, v V.

4 4 BURMAN, GUZMÁN, AND LEYKEKHMAN Figure 1. Sketc of 0 and + s 1.2. Main Result. Te main goal of tis paper is to obtain weigted a priori error estimates, were te weigt ω is O1 on =, x 0 ] [y 1, y 2 ] and decays exponentially outside of a sligtly larger subdomain cf. Figure 1. More precisely, te weigt ω is a positive function wit te following properties: C 1 ωx, y C 2, for x, y 0, ωx, y Ce x x0/k, for x x 0 +, ωx, y Ce y y2/kσ, for y y 2 +, ωx, y Ce y1 y/kσ, for y y 1. Here C 1 and C 2 are two fixed positive constants, K > 1 is a sufficiently large number, and σ, te size of te crosswind layer, is 1.5 σ := maxε 1/2, 3/4. Teorem 1.1. For every u H 1 and U V tat satisfy 1.3 wit ε, tere exists a constant C independent of u and U, suc tat 1/2 ωu U x L2 + 3/4 ωu U y L2 + Qu U C χ V Lu χ, were Q 2 v :=ε ω v 2 L ω ω x 1/2 v 2 L 2 + ωv 2 L ω[v x ] 2 L 2 e + ω[v y] 2 L 2 e e E ωv n x 1/2 2 L 2 + 1/2 ωvn y 2 L 2 + γ bcε 1 ωv 2 L 2

5 WEIGHTED ESTIMATES OF CIP METHOD 5 and L 2 v := ω v 2 L ωv 2 L e E ω[v x ] 2 L 2 e + ω[v y] 2 L 2 e. We point out tat by penalizing te crosswind derivative i.e. by including te term J in te metod we reduce te size of te crosswind layer for piecewise linear elements to order σ = max 3/4, ε 1/2 instead of order σ = 1/2. To give an application of te above result, let 0 be as in 1.4 and define + s = {x x 0 + sk log, y 1 skσ log y y 2 + skσ log }. Corollary 1.2. Under te assumptions of Teorem 1.1 and assug u H t, tere exists a constant C independent of u and suc tat for any s > 0, wit r u = 2, t. 1/2 u U x L /4 u U y L u U L 2 0 C ru 1 2 u H ru + s + Cs+ru 1 2 u H ru, Remark 1. Note tat in te above estimate r u can be cosen to take different values in te different subdomains in te presence of singularities. Te rest of te paper is organized as follows. Te next tree sections are devoted to te proof of Teorem 1.1 for te piecewise linear case and constitute te main part of te paper. In Section 2 and Section 3, we collect some preliary results wic are necessary in order to carry out te proof of te Teorem 1.1. Te proofs of Teorem 1.1 and Corollary 1.2 are presented in Section 4. Te last section addresses te possible extensions, generalizations, and concluding remarks. Finally in te Appendix, we provide te proofs of te tecnical results stated in Section Preliary Results In tis section we collect some results we require in our analysis. First we recall te standard trace and inverse inequalities. Te proofs can be found in many textbooks on finite elements cf. [6] Trace, inverse and interpolation inequalities. For T T and v H 1 T we ave 2.1 v L 2 T C tr 1/2 v L 2 T + 1/2 v L 2 T, were C tr is independent of T and v. If v P 1 T, ten 2.2 v L2 T C inv 1 v L2 T, 2.3 v L 2 T C inv 1/2 v L 2 T, were C inv is independent of and v. Let T T and v H 2 T, ten te interpolation inequality reads 2.4 v L2 T C 1 v L2 T + D 2 v L2 T, for some constant C independent of T and v.

6 6 BURMAN, GUZMÁN, AND LEYKEKHMAN 2.2. Te weigt function. In addition to te properties ω described above, we assume tat ω satisfies, ω x x, y < 0, for all x, y, Dx α Dy β ω CK α α K β σ β ω, for α + β 2, Dx α Dy β ω CK 1 α 1 α K β σ β ω x, for α 1, α + β 2, ROS, ω + ROS, ω x C ω, for any ball S of radius K, were ROS, v = max x S vx / x S vx. Te explicit construction of suc a function is given in [14]. Te last property of ω enables us to apply te inverse inequalities 2.2 and 2.3 to functions of te form ωv, wit v P 1 T. For example, 2.9 and similarly ω v L 2 T max x T ωx v L 2 T max x T ωxc inv 1 v L 2 T C inv C ω x T ωx 1 v L2 T C inv C ω 1 ωv L2 T, 2.10 ωv L 2 T C inv C ω 1/2 ωv L 2 T. Finally, we would like to note tat in te subsequent analysis it is important tat ω 1 = 1 ω as te same properties as ω. For example, 2.11 D x ω 1 = ω 2 D x ω ω 2 CK 1 1 ω = CK 1 1 ω 1 and for any ball of radius K, x T ωx 1 max x T ωx 2.12 ROS, ω 1 = max x T [ωx] 1 1 x T [ωx] 1 = = ROS, ω C ω Quasi-interpolant operator. Next we introduce a quasi-interpolant operator wic is similar to te Clément operator [7]. Te properties of tis interpolant are essential to carry out te error analysis of te CIP metod; see [2, 3, 4]. In tis paper, we use te quasi-interpolant Π presented in [4]. For simplicity we only consider te piecewise linear case. Te value of Πv at a node x i of our finite element space V i.e. te nodes defined by te Lagrangian polynomial basis is given by 2.13 Πvx i := 1 N i {T :x i T } v T x i. Here N i is te number of triangles in T tat contain x i. One of te properties of Π we will use in te subsequent analysis is te following L 2 -stability result 2.14 Πv L2 τ C v L2 τ, v L 2 τ. 3. Tecnical results In tis section we state several results in te form of lemmas, wic we require in order te carry out te proof of te main result. Te proofs of tese lemmas are quite tecnical and are given in te Appendix.

7 WEIGHTED ESTIMATES OF CIP METHOD Weigted estimates for a quasi-interpolant operator. Te first result is a weigted version of Lemma 3.1 in [4]. Tis result is important in our presentation and will be used often. Tis type of estimate is also used in te a posteriori error estimation for non-conforg finite element metods and can be considered to be of independent interest. Lemma 3.1. Let U V and te operator Π be given by Tere exists a constant C independent of U,, and K suc tat, 3.1 ωu x ΠU x 2 L 2 C e E ω 1 ω 2 U x Πω 2 U x 2 L 2 C e E 0 and ω 1 ω 2 U x Πω 2 U x 2 L 2 T C e E 0 ω[u x ] 2 L 2 e, ω[u x ] 2 L 2 e + CK 2 ωu x 2 L 2, ω[u x ] 2 L 2 e + CK 2 ωu x 2 L Estimates for te crosswind derivative. Te next result is special for te piecewise linear case. Tis lemma allows us to prove tat in piecewise linear case te size of te crosswind layer is of order σ = max 3/4, ε 1/2. We would like to point out tat te result below olds true for any function in V and does not use te fact tat U is te approximate solution. Lemma 3.2. Let U V. Tere exists a constant C independent of U and, suc tat 3/2 ωu y 2 L 2 C 2 ω[u y ] 2 L 2 e + ωu 2 L 2 + 1/2 ωun y 2 L 2. e E Estimates for te streamline derivative. In te next result we will give an estimate for te upwind term. In contrast to te above result, tis lemma makes a strong use of te fact tat U is an approximate solution to 1.1. One of te main tecnical tools used to prove te following lemma is te weigted quasi-interpolant results in Lemma 3.1. Lemma 3.3. Let U V and u H 1 satisfy 1.3. Tere is a constant C independent of U, u, and, suc tat ωu x 2 L 2 C Q 2 U + L 2 u Superapproximation. Te following superapproximation result is similar to te superapproximation result in [14] and [12]. Te difference from te above mentioned superapproximation results is tat instead of a local interpolant operator we ere use a global L 2 -projection. Because of tis, te lemma as an independent interest. Te ortogonal properties of te L 2 -projection will be essential in our proof of te teorem. Indeed, tis is wat allows us to control te convective term by te gradient jump operator.

8 8 BURMAN, GUZMÁN, AND LEYKEKHMAN Lemma 3.4. Let U V and u H 1 satisfy 1.3. Let P : L 2 V denote te L 2 -projection defined by 3.4 P v χ dx = v χ dx, χ V. Set E = ω 2 U P ω 2 U. Tere exists a constant C independent of U, u,, and K, suc tat ω 1 E 2 L 2 + ω 1 E 2 L 2 CK 1 Q 2 U + L 2 u. Remark 2. By using te trace inequality 2.1 and property of ω 2.8, we may include ω 1 E L2 on te left and side of Lemma 3.4, since, ω 1 E 2 L 2 x T [ωx] 2 E 2 L 2 T Ctr 2 x T [ωx] 2 1 E 2 L 2 T + E 2 L 2 T CtrC 2 ω 2 1 ω 1 E 2 L 2 + ω 1 E 2 L 2. Te last result states an important inequality wic we will use in te second part of te proof of Teorem 1.1. Te main tecnical tool used to prove tis lemma is te above superappoximation result, Lemma 3.4. Lemma 3.5. Let U V and u H 1 satisfy 1.3. Tere exist constants C and δ, te latter may be cosen arbitrary small, independent of U, u,, and K, and a constant C δ, wic depends on δ, but not on U, u,, and K, suc tat BU, ω 2 U CK 1 Q 2 U + CδQ 2 U + ωu x 2 L 2 + C δl 2 u. 4. Proof of Teorem 1.1 and Corollary 1.2 After collecting all te necessary tecnical results in te previous section, we are ready to present a proof of Teorem 1.1. We prove te teorem in several steps. Proof. Step 1: Reduction to weigted stability. In tis step we will prove tat it is sufficient to sow tat 4.1 QU C δ Lu + Cδ 1/2 ωu x L 2, for all U V and u H 1 tat satisfy 1.3. Here δ is a constant tat can be made as small as required. By te triangle inequality, we ave 1/2 ωu u x L2 + 3/4 ωu u y L2 + QU u 1/2 ωu χ x L2 + 3/4 ωu χ y L2 + QU χ + CLu χ. We ave used tat 1/2 ωu χ x L2 + 3/4 ωu χ y L2 + Qu χ CLu χ, wic follows from te properties of ω, te assumption ε, and te trace inequality Hence, it is enoug to sow tat for any χ V, 1/2 ωu χ x L 2 + 3/4 ωu χ y L 2 + QU χ CLu χ.

9 WEIGHTED ESTIMATES OF CIP METHOD 9 We note tat if χ V, ten Ũ := U χ V, ũ := u χ H 1 and Terefore, it is enoug to sow tat B Ũ ũ, v = 0, v V. 1/2 ωũx L2 + 3/4 ωũy L2 + QŨ CLũ. In view of Lemma 3.2, 3/4 ωũy L2 CQŨ. Moreover, using Lemma 3.3, it is sufficient to sow 4.2 QŨ C δlũ + Cδ 1/2 ωũx L 2, were δ is a constant tat can be made as small as desired. Using U and u instead of Ũ and ũ we see tat we need to sow 4.1 for all U V and u H 1 tat satisfy 1.3. Step 2: Relating Q 2 U to BU, ω 2 U. In tis step we will sow tat 4.3 Q 2 U = BU, ω 2 U 2ε Uω ω U 2ε U n ω2 U + U 2 ω ω n ds. Recalling te definition 1.2, we ave BU, ω 2 U = εau, ω 2 U + MU, ω 2 U + J U, ω 2 U + J U, ω 2 U = ε ω U 2 L 2 + 2ε Uω ω U U ε n ω2 U + ω2 U n U ds + εγ bc ωu 2 L 2 + U x ω 2 U + ωu 2 L 2 + ω 2 U 2 n x ds [Ux ][ω 2 U x ] + [U y ][ω 2 U y ] ds + 1/2 ωu n y 2 L e E 0 e Let us first treat te first term in MU, ω 2 U, namely U xω 2 U. By integration by parts and using tat ω x < 0, we ave U x ω 2 U = U 2 ωω x + 1 ω 2 U 2 n x ds 2 = ω ω x 1/2 U 2 L ω 2 U 2 n x ds + 1 ω 2 U 2 n x ds. 2 2 \ Because n x < 0 on, we ave ω 2 U 2 n x ds + 1 ω 2 U 2 n x ds + 1 ω 2 U 2 n x ds 2 2 \ = 1 ω 2 U 2 n x ds + 1 ω 2 U 2 n x ds 2 2 \ = 1 2 ωu n x 1/2 2 L 2.

10 10 BURMAN, GUZMÁN, AND LEYKEKHMAN Since, ω is smoot and U is continuous, te jump terms [ω 2 x U] and [ω 2 y U] vanis and we ave [Ux ][ω 2 U x ] + [U y ][ω 2 U y ] ds e = [Ux ][ω 2 x U + ω 2 U x ] + [U y ][ω 2 y U + ω 2 U y ] ds e = ω[u x ] 2 L 2 e + ω[u y] 2 L 2 e. Taking into account te above arguments, we ave sown 4.3. Step 3: Initial Estimate for Q 2 U. In tis step we will bound te last two terms on te rigt and side of 4.3 to sow 4.4 Q 2 U CBU, ω 2 U. For te second term appearing in te rigt and side of 4.3 we note tat Uω ω U = Uωω x U x + ω y U y, ence by te Caucy-Scwarz inequality, 2.6, and using tat ε, ε Uωω x U x ε ωu x L 2 ω x U L ε ωu x L 2 C 1/2 K 1/2 ω ω x 1/2 U L 2 CK 1/2 ε ω U 2 L 2 + ε 1 ω ω x 1/2 U 2 L 2 CK 1/2 Q 2 U. Similarly, by 2.6 and using tat by definition 1.5 σ = max ε 1/2, 3/4 and terefore ε σ 2, 4.6 ε ωuω y U y ε ωu y L2 ω y U L2 ε ωu y L2 Cσ 1 K 1 ωu L2 CK 1 ε ω U 2 L 2 + εσ 2 ωu 2 L 2 CK 1 Q 2 U. Combining, 4.5 and 4.6, we obtain 4.7 ε ωu ω U CK 1/2 Q 2 U. Next we will estimate te boundary term. By te Caucy-Scwarz inequality, te property 2.8 of ω, and te trace inequality 2.10, U ε n ω2 U ds ε ω U n L 2 ωu L2 εc ω C inv 1/2 ω U L2 ωu L2 ε 4 ω U 2 L 2 + ωc 2 εc2 inv ωu 2 L 2. Also, using 2.6 and 2.8, we see tat ε U 2 ω ω n ds ε CK 1 ωu 2 L 2.

11 WEIGHTED ESTIMATES OF CIP METHOD 11 Terefore, if γ bc > 2C 2 inv C2 ω and by taking K large enoug, we ave U 4.8 2ε n ω2 U + U 2 ω ω ds 1 n 2 Q2 U. If we substitute 4.8 and 4.7 into 4.3 and coose K sufficienlty large we ave 4.4. Step 4: Final estimate for Q 2 U. Applying Lemma 3.5 to te rigt and side of 4.4, we obtain Q 2 U CK 1 Q 2 U + CδQ 2 U + ωu x 2 L 2 + C δl 2 u. Coosing K large and δ small, sows 4.1. Tis proves te teorem. Now we will prove Corollary 1.2 Proof. By Teorem 1.1 and te triangle inequality, for any χ V we ave, 1/2 u U x L /4 u U y L2 0 + u U L2 0 C 1/2 ωu U x L 2 + 3/4 ωu U y L 2 + ωu U L 2 C Lu χ C 1/2 u χ L 2 + s + 1/2 u χ L 2 + s + + 1/2 ω u χ L 2 \ + s + 1/2 ωu χ L 2 \ + s 1/2 + ω u χ 2 L 2 e. e E \ + s e E + s u χ 2 L 2 e Taking χ = I u to be te Lagrange interpolant of u and using te approximation teory togeter wit te trace inequality 2.1, we obtain 1 2 1/2 u I u L 2 + s + 1/2 u I u L 2 + s + u I u 2 L 2 e 1 2 C ru 1 2 u H ru + s. e E + s For te oter terms using tat ω = O s outside of + s, we ave 1/2 ω u I u L2 \ + s + 1/2 ωu I u L2 \ + s 1/2 + ω u I u 2 L e 2 e E \ + s 1/2 C s 1/2 u I u L2 + 1/2 u I u L2 + u I u 2 L 2 e. e E

12 12 BURMAN, GUZMÁN, AND LEYKEKHMAN Similarly, by te trace inequality 2.1 and te approximation teory, we ave 1/2 u I u L 2 + 1/2 u I u L 2 + u I u 2 L 2 e e E C 1/2 u I u L 2 + 1/2 u I u L 2 + ru 1 2 D r u u L C ru 1 2 u H ru. Tis proves te corollary. 5. Extensions and concluding remarks Tere are several extensions of te above analysis tat are straigtforward altoug tecnical. Below we will comment on some of te more important ones Hig-order elements. Te extension of te present analysis to ig-order elements is straigtforward. In our analysis, only Lemma 3.2 makes strong use of te vanising second derivative in an essential way. It seems tat te crosswind jump term i.e. J is useful to control crosswind smearing only in te piecewise linear case. It remains unclear weter te crosswind stabilization for ig-order elements as any effect. However, we can still prove a sligtly weaker result tan Teorem 1.1 of te form 5.1 1/2 ωu U x L 2 + Qu U C χ V Lu χ, were we need to take σ = 1/2. Tis means tat we can only prove tat te crosswind layer is of size 1/2 rater tan max 3/4, ε 1/2. It is not clear if tis result is sarp for ig-order elements. Since te Lemma 3.2 does not old for k 2, in order to establis 5.1 one will need a sligtly different superapproximation result of te form ω 1 E 2 L 2 + ω 1 E 2 L 2 CK 1 ωu 2 L 2 + ωω x 1/2 U 2 L L -norm estimates. In tis paper we ave presented weigted error estimates in L 2 -norm. Using similar weigted stability estimates one can prove suboptimal L -norm estimates in regions away from boundary layers. More precisely, if we use te tecnique in [15], we can prove u U L 0 C 11/8, were 0 and te distance from 0 to te outflow part of te boundary of is at least K log1/ for a sufficiently large constant K No reaction term and variable convection. In our presentation, for simplicity, we assumed te reaction term in 1.1 is just u. However, if instead we ave used te weigt function presented in [12, 16], we could ave sown te results for a more general problem 5.3 ε u + β u + cu = f in, u = 0 were β [L ] 2 and c L, cx 0. on,

13 WEIGHTED ESTIMATES OF CIP METHOD Summary and concluding remarks. We ave proved a weigted a priori error estimate for te CIP metod applied to a convection doated second order convection-diffusion equation. We also proved tat by including extra terms we can reduce te size of numerical layer in te crosswind direction. Typically te numerical layer in te upwind direction is of order O log 1/ wereas it is of order O 1/2 log 1/ in te crosswind direction. In our analysis of te piecewise linear case by adding an extra penalty term and using σ = max 3/4, ε 1/2, we are able to reduce te size of te crosswind numerical layer to order O 3/4 log 1/. Te reason we were able to accomplis tis is tat once Lemma 3.2 was establised, we could use te superapproximation result, Lemma 3.4. Te reason tat Lemma 3.2 olds is tat we ave included te term J in te definition of te CIP metod. In oter words, penalizing te jumps of te crosswind derivative allows us to reduce te size of te crosswind numerical layer to σ = maxε 1/2, 3/4 in te case of piecewise linear approximations. Tis argument is similar to te results of [14], were more crosswind diffusion was added in order to reduce te crosswind smear. Note owever tat in spite of te fact tat te CIP-crosswind diffusion is weakly consistent tis property does not seem to generalize to te case of iger order polynomial approximation. Tese types of results are expected to be of interest in problems in fluid mecanics. For example, if similar results eld for te CIP-metod of [4] applied to te Oseen s equation it would mean tat, away from boundary layers and singularities, solutions to te linearized equations of incompressible flow are accurately approximated by te CIP-metod also in te ig Reynolds number regime Proof of Lemma Appendix: Tecnical proofs Proof. Let T denote te closure of te set of triangles T T suc tat T T. We start by proving 3.1. By [4], for any piecewise polynomial function p 6.1 p Πp 2 L 2 T C [p] 2 L 2 e. e E 0,e T Using 2.8 and 6.1, we can estimate te left and side of 3.1 as ωu x ΠU x 2 L 2 max x T [ωx]2 U x ΠU x 2 L 2 T C max x T [ωx]2 [U x ] 2 L 2 e e E 0,e T CC 2 ω e E 0 ω[u x ] 2 L 2 e, wic proves 3.1. Next we will establis 3.2. By te triangle inequality, ω 1 ω 2 U x Πω 2 U x 2 L 2 C ω 1 Πω 2 U x Πω 2 U x 2 L 2 + ω 1 Πω 2 U x ω 2 U x 2 L 2 + ω 1 ω 2 U x ω 2 U x 2 L 2 = CI 1 + I 2 + I 3,

14 14 BURMAN, GUZMÁN, AND LEYKEKHMAN were ω 2 = 1 T T ω2 is te average of ω 2 over eac triangle. Using 2.12 and te stability of Π in te L 2 -norm, we ave 6.2 I 1 = ω 1 Πω 2 U x Πω 2 U x 2 L 2 T C x T [ωx] 2 ω 2 U x ω 2 U x 2 L 2 T CC 2 ω ω 1 ω 2 U x ω 2 U x 2 L 2 = CC2 ωi 3. Tus, we need only to estimate I 2 and I 3. To estimate I 3 we use te fact tat ω 2 ω 2 L T C ω 2 L T and te properties of ω 2.6 and Tus, 6.3 I 3 = ω 1 ω 2 U x ω 2 U x 2 L 2 T C x T [ωx] 2 2 ω 2 2 L T U x 2 L 2 T C x T [ωx] 2 K 2 ω 4 L T U x 2 L 2 T CCω 4 K 2 ωu x 2 L 2 T = CC4 ω K 2 ωu x 2 L 2. Tus, we are only left to estimate I 2. Again using 2.12, 6.1, and te triangle inequality, we ave 6.4 I 2 = ω 1 Πω 2 U x ω 2 U x 2 L 2 T x T [ωx] 2 Πω 2 U x ω 2 U x 2 L 2 T C x T [ωx] 2 C x T [ωx] 2 [ω 2 U x ] 2 L 2 e e E 0,e T e E 0,e T [ω 2 U x ] 2 L 2 e + [ω2 ω 2 U x ] 2 L 2 e. Since ω is smoot, [ω 2 U x ] = ω 2 [U x ], and using 2.12, we ave x T [ωx] 2 ω 2 [U x ] 2 L 2 e C2 ω ω[u x ] 2 L 2 e e E 0,e T e E 0,e T CCω 2 ω[u x ] 2 L 2 e. e E 0

15 WEIGHTED ESTIMATES OF CIP METHOD 15 To estimate te remaining term in I 2, we proceed similarly to 6.3 and use te inverse inequality 2.3 applied to [U x ], x T [ωx] 2 [ω 2 ω 2 U x ] 2 L 2 e e E 0,e T C x T [ωx] 2 2 ω 2 2 L T [U x ] 2 L 2 e e E 0,e T CCinv 2 x T [ωx] 2 K 2 ω 4 L T 1 U x 2 L 2 T T T CCinvC 2 ωk 4 2 ωu x 2 L 2 T CC 2 invc 4 ωk 2 T T ωu x 2 L 2 T = CC2 invc 4 ωk 2 ωu x 2 L 2. Combining te above two estimates wit estimates 6.2, 6.4, and 6.3, we prove 3.2. Finally, to obtain 3.3, we apply 2.12, triangle inequality, and te inverse inequality to ω 2 U x Πω 2 U x, we obtain, 2 ω 1 ω 2 U x Πω 2 U x 2 L 2 T C 2 x T [ωx] 2 ω 2 U x Πω 2 U x 2 L 2 T C 2 x T [ωx] 2 ω 2 U x ω 2 U x 2 L 2 T + ω2 U x Πω 2 U x 2 L 2 T C x T [ωx] 2 2 ω 2 U x ω 2 U x 2 L 2 T + C2 inv ω 2 U x Πω 2 U x 2 L 2 T CCω 2 2 ω 1 ω 2 ω 2 U x 2 L 2 T + CC2 invcω 2 ω 1 ω 2 U x Πω 2 U x 2 L 2 T CCω 2 2 ω 1 ω 2 ω 2 U x 2 L 2 T + CC2 invc 2 ωi 1 + I 2. Since we already sowed te desired estimate for I 1 and I 2, to finis te proof we use 2.6, 2.12, and noticing tat U x = 0, we obtain, 2 ω 1 ω 2 ω 2 U x 2 L 2 T = 2 2 ωu x 2 L 2 T CK 2 ωu x 2 L 2 T. Tis concludes te proof of te lemma Proof of Lemma 3.2. Proof. Using te integration by parts and te fact tat U y is piecewise constant, 6.5 ωu y 2 L 2 = T ω 2 U 2 y = 2ωω y UU y + ω 2 UU y n y ds. T T

16 16 BURMAN, GUZMÁN, AND LEYKEKHMAN Using te property of ω, namely 2.6, te Caucy-Scwarz and aritmetic-geometric mean inequalities, we can estimate te first term on te rigt and side by 6.6 3/2 T 2ωω y UU y C 3/4 K 1 ωu y L 2 ωu L 2 3/2 4 ωu y 2 L 2 + CK 2 ωu 2 L 2. By te Caucy-Scwarz and aritmetic-geometric mean inequalities, we can estimate te second term on te rigt and side by 3/2 Since by 2.10 we obtain 6.7 T ω 2 UU y n y ds 3/2 e E 0 ω[u y ] L2 e ωu L2 e + 3/2 ωun y L 2 ωu y L ω[u y ] 2 L 2 2 e + ωu 2 L 2 e e E 0 ωu y 2 L 2 = ωu y 2 L 2 e 3/2 e E C 2 invc 2 ω 1 T + 5/2 4Cinv 2 ωu y 2 C2 L 2 + C1/2 ωun y 2 L 2. ω ωu y 2 L 2 T ωu y 2 L 2 T = C2 invc 2 ω 1 ωu y 2 L 2, ω 2 UU y n y ds C 2 e E 0 ω[u y ] 2 L 2 e + C ωu 2 L 2 + 3/2 4 ωu y 2 L 2 + C1/2 ωun y 2 L 2. By absorbing te term 3/2 4 ωu y 2 L 2 appearing in 6.6 and 6.7 by te left and side of 6.5, we complete te proof of te lemma Proof of Lemma 3.3. Proof. We start te proof of te lemma by adding and subtracting te Clément interpolant of ω 2 U x, 6.8 ωu x 2 L 2 = U x Πω 2 U x + U x ω 2 U x Πω 2 U x.

17 WEIGHTED ESTIMATES OF CIP METHOD 17 Using te Caucy-Scwarz and geometric-aritmetic mean inequalities and te Lemma 3.1, we can estimate te last term on te rigt and side by U x ω 2 U x Πω 2 U x ωu x L 2 ω 1 ω 2 U x Πω 2 U x L ωu x 2 L 2 + ω 1 ω 2 U x Πω 2 U x 2 L ωu x 2 L 2 + CK 2 ωu x 2 L 2 + C e E 0 ω[u x ] 2 L 2 e. By coosing K large enoug we can kick back te first two terms on te rigt and side and from 6.8 we obtain ωu x 2 L 2 2 U x Πω 2 U x + C e E 0 ω[u x ] 2 L 2 e. Since 2 e E ω[u 0 x ] 2 L 2 e is one of te terms of Q2 U, to complete te proof, we sall sow 6.9 U x Πω 2 U x 4 ωu x 2 L 2 + C Q 2 U + L 2 u. To establis 6.9 we will use te ortogonality condition 1.3, 6.10 BU, Πω 2 U x = Bu, Πω 2 U x. Tus from 6.10, 6.11 U x Πω 2 U x = Bu, Πω 2 U x U Πω 2 U x n x ds U Πω 2 U x εau, Πω 2 U x J U, Πω 2 U x J U, Πω 2 U x. We bound eac term of te rigt and side separately. In details we will only demonstrate te estimates for εau, Πω 2 U x and J U, Πω 2 U x. Te estimates for te oter terms are very similar. We start wit 6.12 εau, Πω 2 U x = ε U Πω 2 U x U + ε n Πω2 U x + Πω2 U x U ds n ε γ bc 1 UΠω 2 U x ds = I 1 + I 2 + I 3. Adding and subtracting ω 2 U x we ave I 1 = ε T U ω 2 U x ε T U Πω 2 U x ω 2 U x.

18 18 BURMAN, GUZMÁN, AND LEYKEKHMAN Te first term on te rigt and side we can estimate by using te properties of te weigt function and te fact tat U x is piecewise constant, ε U ω 2 U x = ε U ω 2 U x 6.13 T T = 2ε U ωωu x CεK 1 1 ω U 2 L 2 T = CεK 1 1 ω U 2 L 2. To estimate te oter term, we use te Caucy-Scwarz and aritmetic-geometric mean inequalities and Lemma 3.1, ε U Πω 2 U x ω 2 U x 6.14 T ε 2 1 T ω U 2 L 2 T + ε 2 ε 2 1 ω U 2 L 2 + Cε e E 0 Cε 1 ω U 2 L 2 + C e E 0 ω 1 Πω 2 U x ω 2 U x 2 L 2 T ω[u x ] 2 L 2 e + Cε 1 K 2 ωu x 2 L 2 ω[u x ] 2 L 2 e. In te last step we used te assumption ε. Tus, from 6.13 and 6.14, we get I 1 C 1 Q 2 U. Next, we will bound te remaining terms of εau, Πω 2 U x, namely I 2 and I 3. By te Caucy-Scwarz inequality I 2 ε ω U L 2 ω 1 Πω 2 U x L 2 + ε ω 1 Πω 2 U x L 2 ωu L 2. By te aritmetic-geometric mean inequality ω U L2 ω 1 Πω 2 U x L2 1 2 ω U 2 L ω 1 Πω 2 U x 2 L 2. Using te properties of ω 2.8 and te inverse inequality 2.3, ω U 2 L 2 e E max x e [ωx]2 U 2 L 2 e Cinv 2 1 max x T [ωx]2 U 2 L 2 T CωC 2 inv 2 1 ω U 2 L 2 T = C2 ωc 2 inv 1 ω U 2 L 2.

19 WEIGHTED ESTIMATES OF CIP METHOD 19 Using 2.12, te inverse inequality 2.3, te triangle inequality, and Lemma 3.1, we ave ω 1 Πω 2 U x 2 L 2 x e [ωx] 2 Πω 2 U x 2 L 2 e 6.15 C 2 inv 1 e E x T [ωx] 2 Πω 2 U x 2 L 2 T Cinv 2 1 x T [ωx] 2 Πω 2 U x ω 2 U x 2 L 2 T + ω2 U x 2 L 2 T CωC 2 inv 2 1 ω 1 Πω 2 U x ω 2 U x 2 L 2 T + ωu x 2 L 2 T C e E 0 C e E 0 ω[u x ] 2 L 2 e + C 1 K 2 ωu x 2 L 2 + C2 ωc 2 inv 1 ωu x 2 L 2 ω[u x ] 2 L 2 e + C 1 ω U 2 L 2. Using te assumption ε, we ave sown, 6.16 ε ω U L2 ω 1 Πω 2 U x L2 C 1 Q 2 U. Similarly, by te aritmetic-geometric mean inequality ω 1 Πω 2 U x L 2 ωu L 2 2 ω 1 Πω 2 U x 2 L ωu 2 L 2. Proceeding exactly as in te estimate 6.15, we obtain 2 ω 1 Πω 2 U x 2 L 2 C e E 0 Again using te assumption ε, we ave ω[u x ] 2 L 2 e + C 1 ω U 2 L ε ω U L 2 ω 1 Πω 2 U x L 2 C 1 Q 2 U. Tus, from 6.16 and 6.17, we ave Similarly, we can sow Tus, we ave sown I 2 C 1 Q 2 U. I 3 C 1 Q 2 U εau, Πω 2 U x CQ 2 U. In a similar fasion, we can sow UΠω 2 U x n x ds + UΠω 2 U x + J U, Πω 2 U x J U, Πω 2 U x CQ2 U + CK 2 + δ ωu x 2 L 2. We will demonstrate tis for J U, Πω 2 U x = 2 [U y ][Πω 2 U x y ]ds + 1/2 e E 0 e UΠω 2 U x n y 2 ds.

20 20 BURMAN, GUZMÁN, AND LEYKEKHMAN We start wit te last term. By te Caucy-Scwarz and aritmetic-geometric mean inequalities, 1/2 UΠω 2 U x n y 2 ds C δ 1/2 ωun y 2 L 2 + δ3/2 ω 1 Πω 2 U x 2 L 2 Since 1/2 ωun y 2 L 2 is one of te terms of Q2 U, we only need to treat te last term. By te properties of ω, te inverse inequality 2.3, te triangle inequality, and Lemma 3.1, ω 1 Πω 2 U x 2 L 2 C2 invc 2 ω 1 ω 1 Πω 2 U x 2 L 2 Tus, 6.20 C 2 invc 2 ω 1 ω 1 Πω 2 U x ω 2 U x 2 L 2 + ωu x 2 L 2 C e E 0 1/2 ω[u x ] 2 L 2 e + 1 CK 2 + C 2 invc 2 ω ωu x 2 L 2. UΠω 2 U x n y 2 ds C δ 1/2 ωun y 2 L 2 + C 3/2 e E 0 ω[u x ] 2 L 2 e + δ1/2 CK 2 + C 2 invc 2 ω ωu x 2 L 2. Next we estimate te oter term of J. By te aritmetic-geometric mean inequality, 2 y ][Πω e[u 2 U x y ]ds C ω[u y ] 2 L 2 e + 3 ω 1 [Πω 2 U x y ] 2 L 2 e. e E 0 e E 0 Again, 2 e E ω[u 0 y ] 2 L 2 e is one of te terms of Q2 U, ence we only need to treat te last term. By te properties of ω, te inverse inequality 2.3, te triangle inequality, and Lemma 3.1, 3 ω 1 [Πω 2 U x y ] 2 L 2 e C2 x T [ωx] 2 Πω 2 U x 2 L 2 T e E 0 C 2 ω 1 Πω 2 U x ω 2 U x 2 L 2 T + ωu x 2 L 2 T Tus, 6.21 C e E 0 2 e E 0 ω[u x ] 2 L 2 e + CK 2 ωu x 2 L 2. e[u y ][Πω 2 U x y ]ds C e E 0 Combining 6.20 and 6.21 we obtain + CK 2 ωu x 2 L 2. ω[u x ] 2 L 2 e + ω[u y] 2 L 2 e J U, Πω 2 U x CQ 2 U + CK 2 + δ 3/2 C 2 invc 2 ω ωux 2 L 2. Terefore multiplying 6.11 by and using 6.18 and 6.19, 6.22 U x Πω 2 U x CQ 2 U + CK 2 + δ ωu x 2 L 2 + Bu, Πω2 U x.

21 WEIGHTED ESTIMATES OF CIP METHOD 21 It remains to bound Bu, Πω 2 U x = εau, Πω 2 U x +Mu, Πω 2 U x +J u, Πω 2 U x +J u, Πω 2 U x. Te bound of te first term εau, Πω 2 U x follows in te same fasion as Now, we bound Mu, Πω 2 U x = u x Πω 2 U x + uπω 2 U x n x ds + uπω 2 U x. Adding and subtracting ω 2 U x, we ave u x Πω 2 U x = u x ω 2 U x + u x Πω 2 U x ω 2 U x. By te aritmetic-geometric mean inequality u x ω 2 U x 1 16 ωu x 2 L 2 + C ωu x 2 L 2, and by Lemma 3.1, u x Πω 2 U x ω 2 U x C ωu x 2 L 2 + C ω 1 Πω 2 U x ω 2 U x 2 L 2 C ωu x 2 L 2 + C 1 Q 2 U + CK 2 ωu x 2 L 2. Terefore, u x Πω 2 U x 16 ωu x 2 L 2 + CL2 u + CQ 2 U + CK 2 ωu x 2 L 2. We can easily bound te remaining terms of Mu, Πω 2 U x to arrive at Mu, Πω 2 U x 8 ωu x 2 L 2 + CL2 u + CQ 2 U + CK 2 ωu x 2 L 2. Te estimates of J u, Πω 2 U x and J u, Πω 2 U x can be derived along te lines of te estimate for J U, Πω 2 U x. Assembling all te bounds on te terms of Bu, Πω 2 U x we obtain Bu, Πω 2 U x 6 ωu x 2 L 2 + CL2 u + CQ 2 U + CK 2 + δ ωu x 2 L 2. Using te above inequality, estimate 6.22, and taking K large enoug and δ small enoug, proves 6.9. Tis completes te proof of te lemma Proof of Lemma 3.4. As we ave already mentioned above, tis superapproximation result is similar to te superapproximation results of [13] and [10], but ere instead of a local interpolant operator we ave to deal wit a global L 2 - projection. Because of tis fact te proof is muc more involved. Proof. Recall tat E = ω 2 U P ω 2 U, were P is te L 2 -projection defined in 3.4. Using tat u P u is ortogonal to V, we ave ω 1 E 2 L 2 = ω 2 ω 2 U P ω 2 Uω 2 U P ω 2 U = ω 2 U P ω 2 UU ω 2 P ω 2 U = ω 2 U P ω 2 UI ω 2 P ω 2 U ω 2 P ω 2 U,

22 22 BURMAN, GUZMÁN, AND LEYKEKHMAN were I denotes te Lagrange interpolant. Tus, by te Caucy-Scwarz inequality, Hence, ω 1 E 2 L 2 ω 1 E L2 ωi ω 2 P ω 2 U ω 2 P ω 2 U L2. ω 1 E L2 ωi ω 2 P ω 2 U ω 2 P ω 2 U L2. By following te proof of Lemma 2.2 in [14] and using 2.11 and 2.12, we get 1 ωi ω 2 P ω 2 U ω 2 P ω 2 U L2 CK 1/2 1/2 1/2 ω 1 P ω 2 U x L 2 + 3/4 ω 1 P ω 2 U y L 2 + ω 1 P ω 2 U L2 + ω 1 ω 1 x 1/2 P ω 2 U L2. Terefore by te triangle inequality, ω 1 E L 2 = CK 1/2 1/2 S 1 + S 2, were and S 1 = 1/2 ω 1 ω 2 U x L 2 + 3/4 ω 1 ω 2 U y L 2 + ωu L 2 + ω 1 ω 1 x 1/2 ω 2 U L 2. S 2 = 1/2 ω 1 P ω 2 U ω 2 U x L 2 + 3/4 ω 1 P ω 2 U ω 2 U y L 2 + ω 1 P ω 2 U ω 2 U L2 + ω 1 ω 1 x 1/2 P ω 2 U ω 2 U L2. One can sow using te product rule and 2.6, 2.7, and 2.8 tat S 1 C 1/2 ωu x L 2 + C 3/4 ωu y L 2 + ωu L 2 + ωω x 1/2 U L 2. Terefore, by Lemma 3.2 and Lemma 3.3, we ave S 1 CQU + Lu. Now we bound S 2. It easily follows tat S2 2 C ω 1 P ω 2 U ω 2 U x 2 L 2 T + 3/2 ω 1 P ω 2 U ω 2 U y 2 L 2 T + ω 1 P ω 2 U ω 2 U 2 L 2 T + ω 1 ω 1 x 1/2 P ω 2 U ω 2 U 2 L 2 T We analyze te first term. By using 2.12 and te interpolation inequality 2.4, we obtain ω 1 P ω 2 U ω 2 U x 2 L 2 T = CCω 2. x T [ωx] 2 P ω 2 U ω 2 U x 2 L 2 T 2 ω 1 P ω 2 U ω 2 U 2 L 2 T + 2 ω 1 D 2 P ω 2 U ω 2 U 2 L 2 T Since P ω 2 U is piecewise linear, ω 1 D 2 P ω 2 U ω 2 U 2 L 2 T = ω 1 D 2 ω 2 U 2 L 2 T,.

23 WEIGHTED ESTIMATES OF CIP METHOD 23 and we can sow using te product rule along wit properties of ω tat 3 ωu x 2 L 2 + 3/2 ωu y 2 L ω 1 D 2 ω 2 U 2 L 2 T C + ωu 2 L 2 + ωω x 1/2 U 2 L 2 Togeter wit Lemma 3.3 and Lemma 3.2, we obtain 3 ω 1 D 2 P ω 2 U ω 2 U 2 L 2 T C Q2 U + L 2 u. Terefore, we ave sown ω 1 P ω 2 U ω 2 U x 2 L 2 T C 1 ω 1 E 2 L 2 + CQ2 U + L 2 u. In a similar manner we can bound te remaining terms of S 2 2 to get S 2 2 C 1 ω 1 E 2 L 2 + C Q2 U + L 2 u. By taking te square root of bot sides we get S 2 C 1/2 ω 1 E L2 + C QU + Lu. Terefore, if we use te bounds for S 1 and S 2, 6.23, we see tat for K large enoug By 2.8 and 2.4, we get 1 ω 1 E L2 C 1/2 K 1/2 QU + Lu. ω 1 E 2 L 2 C 2 ω 1 E 2 L 2 + C2 Te proof is complete once we use te estimate ω 1 D 2 ω 2 U L Proof of Lemma 3.5. Presenting te proof of tis lemma, we assume tat te reader is already familiar wit proofs of te previous lemmas. Hence, in te proof below, we skip some steps wic appeared already several times in te proofs of te previous lemmas. Proof. By adding and subtracting BU, P ω 2 U and using te ortogonality property 1.3, we ave 6.25 BU, ω 2 U = BU, ω 2 U P ω 2 U + BU, P ω 2 U = BU, E + Bu, P ω 2 U, wit E = ω 2 U P ω 2 U, were P is te L 2 -projection defined in 3.4. First we bound BU, E. Recall tat We start wit εau, E = ε BU, E = εau, E + MU, E + J U, E + J U, E.. U E ε U n E + E n U ds + γ bc ε UE ds.

24 24 BURMAN, GUZMÁN, AND LEYKEKHMAN Te first term can be bounded by using Lemma 3.4 and te assumption ε as follows: ε U E ε ω U L 2 ω 1 E L 2 C ε 1/2 K 1/2 ω U L 2 QU + Lu CK 1/2 ε ω U 2 L 2 + CK 1/2 Q 2 U + L 2 u CK 1/2 Q 2 U + L 2 u. Te remaining terms of εau, E can be bounded in a similar way. Tus we get εau, E CK 1/2 Q 2 U + L 2 u. Te next term we will treat is MU, E = U x E + UE n x ds + UE. By using tat E is ortogonal to V, te Caucy-Scwarz inequality, Lemma 3.1 and Lemma 3.4, U x E = U x ΠU x E C 1/2 ωu x ΠU x L2 1/2 ω 1 E L2 C e E 0 2 ω[u x ] 2 L 2 e + 1 ω 1 E 2 L 2 CK 1 Q 2 U + CL 2 u. Similarly, we can bound te last two terms of MU, E. Tus, we obtain MU, E CK 1/2 Q 2 U + CL 2 u. In a similar fasion we may bound te remaining terms of BU, E following te proof of Lemma 3.3 to get 6.26 BU, E CK 1/2 Q 2 U + CL 2 u. It remains to estimate Bu, P ω 2 U. We start wit u εau, P ω 2 U =ε u P ω 2 U ε n P ω2 U + P ω2 U u ds n + εγ bc 1 up ω 2 U ds. Using te Caucy-Scwarz, aritmetic-geometric mean, and te triangle inequalities, we can bound te first term on te rigt and side as follows: ε u P ω 2 U Cε ω u L 2 ω 1 P ω 2 U L 2 C δ ε ω u 2 L 2 ω + εδ 1 E 2 L 2 + ω 1 ω 2 U 2 L 2, were δ is some small number to be cosen later. By te superapproximation result, Lemma 3.4, we ave ω 1 E 2 L 2 C 1 K 1 Q 2 U + L 2 u,

25 WEIGHTED ESTIMATES OF CIP METHOD 25 and by te triangle inequality ω 1 ω 2 U 2 L 2 ω U 2 L ωu 2 L 2. By te properties of te weigt function 2.6 and 2.7, we ave 6.27 ωu 2 L 2 = ωxu ωyu 2 2 C 1 K 1 ω ω x U 2 + Cσ 2 K 2 ω 2 U 2. Using tat ε and ε σ 2 we ave ε u, P ω 2 U C δ ε ω u 2 L 2 + CδK 1 Q 2 U + L 2 u + CδK 1 ω ω x 1/2 U 2 L 2 + CδK 2 ωu 2 L 2 + Cδε ω U 2 L 2 C δ ε ω u 2 L 2 + CδK 1 Q 2 U + L 2 u. Now we will treat te εγ bc 1 up ω2 U ds term. By te aritmetic-geometric mean and triangle inequalities, we ave up ω 2 U ds C δ ωu 2 L 2 + δ ω 1 E 2 L 2 + δ ωu 2 L 2. By te trace inequality 2.1 ωu 2 L 2 C 1 ωu 2 L 2 + C ωu 2 L 2 C 1 ωu 2 L 2 ωu + C 2 L 2 + ω u 2 L 2. Using 2.7, we ave ωu 2 L 2 C 2 K 2 ωu 2 L 2. Tus, 6.28 ωu 2 L 2 C 1 ωu 2 L 2 + ω u 2 L 2 CL 2 u. By te Remark 2, Tus, εγ bc 1 ω 1 E 2 L 2 CK 1 Q 2 U + L 2 u. up ω 2 U ds C δ L 2 u + CK 1 Q 2 U + CδQ 2 U. Next we bound J u, P ω 2 U = 2 [u x ][P ω 2 U x ] ds. e E 0 e By te Caucy-Scwarz and te triangle inequalities 2 [u x ][P ω 2 U x ] ds C δ 2 ω[ u] 2 L 2 e + δ2 ω 1 [P ω 2 U x ] 2 L 2 e. e By 2.12, te inverse inequality 2.3, and te triangle inequality, we ave 2 ω 1 [P ω 2 U x ] 2 L 2 e C ω 1 P ω 2 U x 2 L 2 T e E 0 C ω 1 ω 2 U x 2 L 2 + ω 1 E x 2 L 2 C ω x U 2 L 2 + ωu x 2 L 2 + ω 1 E x 2 L 2

26 26 BURMAN, GUZMÁN, AND LEYKEKHMAN Since ω x U 2 L 2 CK 1 ω ω x 1/2 U 2 L 2, by Lemma 3.3 and Lemma 3.4, we ave 2 ω 1 [P ω 2 U x ] 2 L 2 e C K 1 Q 2 U + Q 2 U + L 2 u. Hence e E 0 J u, P ω 2 U C δq 2 U + L 2 u. Te estimate of J u, P ω 2 U is similar. It remains to bound Mu, P ω 2 U = Mu, E + Mu, ω 2 U Te first term can be controlled by using te Caucy-Scwarz inequality and te superapproximation result of Lemma 3.4. Te second term we integrate by parts and split te term in te following fasion Mu, ω 2 U = uω x ωu + uω 2 U x 1 2 ωu L 2 ω ω x 1 2 U L ωu L ωux L 2. Similarly to te analysis above, we obtain, Tus, Mu, ω 2 U δq 2 U + ωu x 2 L 2 + C δl 2 u Bu, P ω 2 U CK 1 Q 2 U + CδQ 2 U + ωu x 2 L 2 + C δl 2 u. Combining estimates 6.28 and 6.29 we conclude te proof of te lemma. Acknowledgments: Te autors would like to tank te anonymous reviewers for very insigtful comments and for elping to improve te presentation of te paper. References [1] R. Becker and M. Braack, Two-level stabilization sceme for te Navier-Stokes equations, Numerical Matematics and Advanced Applications, , Springer, Berlin, [2] E. Burman and P. Hansbo, Edge stabilization for Galerkin approximations of convectiondiffusion-reaction problems, Comput. Metods Appl. Mec. Engrg , [3] E. Burman, A unified analysis of conforg and non-conforg stabilized finite element metods using interior penalty, SIAM J. Num. Anal , [4] E. Burman, M. Fernández, and P. Hansbo, Continuous Interior Penalty for Oseen s Equations, SIAM J. Num. Anal , [5] E. Burman and A. Ern, Continuous interior penalty p-finite element metods for advection and advection-diffusion equations, Mat. Comp , [6] P.G. Ciarlet, Te Finite Element Metod for Elliptic Problems, Nort-Holland, Amsterdam, [7] P. Clément, Approximation by finite elements functions using local regularization, RAIRO Anal. Numer , [8] R. Codina and J. Blasco, Analysis of a stabilized finite element approximation of te transient convection-diffusion-reaction equation using ortogonal subscales. Comput. Vis. Sci , no. 3, [9] J. Douglas and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin metods In Computing metods in applied sciences Second Internat. Sympos., Versailles, 1975, pages Lecture Notes in Pys., Vol. 58. Springer, Berlin, [10] P. Grisvard, Singularities in Boundary Value Problems, Recerces en Matmatiques Appliques [Researc in Applied Matematics], 22. Masson, Paris; Springer-Verlag, Berlin, 1992.

27 WEIGHTED ESTIMATES OF CIP METHOD 27 [11] J.-L. Guermond. Stabilization of Galerkin approximations of transport equations by subgrid modeling. M2AN Mat. Model. Numer. Anal , no. 6, [12] J. Guzmán, Local analysis of discontinuous Galerkin metods applied to singularly perturbed problems, J. Numer. Mat , [13] C. Jonson, U. Nävert, J. Pitkänta, Finite element metods for linear yperbolic problems, Comput. Metods Appl. Mec. Engrg , [14] C. Jonson, A.H. Scatz, and L.B. Walbin, Crosswind smear and pointwise errors in streamline diffusion finite element metods, Mat. Comp , [15] K. Niijima, Pointwise error estimates for a streamline diffusion finite element sceme, Numer. Mat , [16] G. Sangalli, Global and local error analysis for te residual-free bubbles metod applied to advection-doated problems, SIAM J. Numer. Anal , Department of Matematics, University of Sussex, Brigton, BN1 9RF UK, E.N.Burman@sussex.ac.uk. Scool of Matematics, University of Minnesota, Minneapolis, MN 55455, USA, guzma033@umn.edu. Department of Matematics, University of Connecticut, Storrs, CT 06269, USA, leykekman@mat.uconn.edu.