Part VIII, Chapter 39. Fluctuation-based stabilization Model problem

Transcription

1 Part VIII, Capter 39 Fluctuation-based stabilization Tis capter presents a unified analysis of recent stabilization tecniques for te standard Galerkin approximation of first-order PDEs using H 1 - conforming finite elements. Te gradient of a function in tis space generally exibits jumps at mes interfaces. Tis means tat only some part of te gradient can be controlled by test functions from te same space, and te remainder part, wic can be viewed as a fluctuation, needs to be controlled by some stabilization tecnique. Tree tecniques are considered erein: Continuous Interior Penalty (CIP) penalizing gradient jumps at mes interfaces, and two closely related tecniques based on a two-scale decomposition of te finite element setting, Local Projection Stabilization (LPS) penalizing a suitable fluctuation of gradients and Subgrid Viscosity (SGV) penalizing te gradient of a suitable fluctuation. Trougout tis capter, boundary conditions are enforced weakly by te boundary penalty tecnique introduced in Model problem We consider te model problem (38.19) wit weakly enforced boundary conditions. Recall tat it consists of finding u V suc tat ã(u,w) = (f,w) L for all w V, wit bilinear form ã(v,w) := (Av,w) L (M N)v,w V,V for all v,w V, were Av = Kv+ d k=1 Ak k v. Assuming tat te fields K and {A k } k {1:d} satisfy (37.1)andtatteboundaryoperatorM satisfies(37.25), tis problem is well-posed, see Proposition In tis capter, we assume tat te exact solution u is in V s = V H s (D;K m ), s > 1 2. Recall tat tis assumption allows us to work wit te boundary fields M,N L ( D;K m m ) and to write ã(v,w) = (Av,w) L ((M N)v,w) L for all v,w V s, wit L = L 2 ( D;K m ).

2 526 Capter 39. Fluctuation-based stabilization 39.2 Stabilized finite element approximation In tis section, we present te finite element setting, te design conditions on te stabilization, and we perform te error analysis. Examples are presented in te next two sections Finite element setting We assume tat we ave at and, for all > 0, an H 1 -conforming finitedimensional space V V built using a sape-regular mes sequence (T ) >0 and a finite element of degree k 1, and a quasi-interpolation operator I : V V wit optimal local approximation properties: Tere is c, uniform wit respect to, suc tat v I (v) L(K) + K (v I (v)) L(K) c r+1 K v H r+1 (D K,K m ), (39.1) for all r [0,k], all v H r+1 (D,K m ), and all K T, were D K collects te mes cells aving a non-empty intersection wit K Local weigts and assumptions on te data Recall from te local quantities β K = max k {1:d} A k L (K;K m m ) for all K T, and te local weigting parameters τ K = ( max(β K 1 K,µ 0) ) 1 = min(β 1 K K,µ 1 0 ), (39.2) were µ 0 results from (37.1c), i.e., µ 0 is associated wit te L-coercivity of te bilinear form ã. In te context of advection-reaction, β K is a local velocity scale, µ 0 scales as te reciprocal of a time, and τ K represents a local time scale. We define te global quantity β D = max K T β K, and we introduce a second local weigting parameter ˇτ K suc tat min(β 1 D K,µ 1 0 ) ˇτ K τ K, K T. (39.3) We will take ˇτ K = min(β 1 D K,µ 1 0 ) for CIP stabilization and ˇτ K = τ K for LPS and SGV stabilization. Wit a sligt abuse of notation, we define te piecewise constant function ˇτ : D R suc tat ˇτ K = ˇτ K for all K T ; te piecewise constant function τ : D R is defined similarly. We additionally assume tat all te fields {A k } k {1:d} are piecewise Lipscitz on a certain partition of D and tat te meses are compatible wit tis partition, so tat te fields {A k K } k {1:d} are Lipscitz for all K T. We denote by L A te largest Lipscitz constant of tese fields. To simplify te tracking of dependencies on model parameters, we assume tat max( K L (D;K m m ), X L (D;K m m ),L A ) c K,X,A µ 0, (39.4) and we ide te factor c K,X,A in te generic constant c used in te error analysis.

3 Part VIII. First-Order PDEs 527 Boundary conditions are enforced using te boundary penalty metod from , i.e., we assume tat tere is a boundary penalty field S L ( D;K m m ) satisfying (38.25) for any boundary face F F, wit ρ F = M F L (F;K m m ). We assume tat tere is a uniform constant c M suc tat ρ F c M β KF for all F F wit F = K F D, see (38.27); for simplicity, we ide te factor c M in te generic constant c used in te error analysis. Our starting point is te following bilinear form on V s V s, see (38.24): ǎ(v,w) = (Av,w) L ((M N)v,w) L +(S v,w) L. (39.5) Design of stabilization and discrete problem Te main idea is to supplement te bilinear form ǎ defined by (39.5) by a stabilization bilinear form s and to consider te following discrete problem: { Find u V suc tat (39.6) a (u,w ) = (f,w ) L, w V, wit a (v,w ) := ǎ(v,w )+s (v,w ). (39.7) To stay somewat general, we only require tat s be defined on V V. Loosely speaking, te purpose of te stabilization bilinear form s is to control te departure of A 1 v from V. We consider te following design requirements on s, were c 1,c 2,c 3 > 0 are uniform wit respect to : (i) s is a symmetric and positive semi-definite, and it satisfies v S := s (v,v ) 1 2 c 1 ˇτ 1 2v L for all v V. (ii) Tere exists a linear map J : V V suc tat, for all v V, c 2 ˇτ 1 2 J (v ) 2 L ˇτ 1 2 A1 v 2 L +µ 0 v 2 L + v 2 S, c 2 ˇτ 1 2 A1 v 2 L (A 1 v,j (v )) L +µ 0 v 2 L + v 2 S. (39.8a) (39.8b) ( )1 (iii) I (v) S c 3 K T (ˇτ 1 K K) 2r+1 K v 2 2 H r+1 (D K;K m ) and all v H r+1 (D;K m ) Error analysis for all r [0,k] We perform te error analysis in te spirit of Strang s First Lemma (see Lemma 19.8). Tis approac is te most general since it does not require us to extend s beyond V V, see Exercise 39.7 for extending s and using Strang s Second Lemma. We consider te space V = V s + V ; since V is H 1 -conforming, V = V s. We define te two following norms on V s :

4 528 Capter 39. Fluctuation-based stabilization v 2 V := µ 0 v 2 L v 2 M + v 2 S + ˇτ 1 2 A1 v 2 L, v 2 V := v 2 V + ˇτ 1 2 v 2 L + ρ 1 2 v 2 L. (39.9a) (39.9b) Te first norm is used to establis inf-sup stability (and well-posedness), and te second one to prove te boundedness of ǎ on V s V. Tese two norms are te same as tose in for GaLS stabilization wit boundary penalty, up to te cange of τ into ˇτ. Lemma 39.1 (Inf-sup stability). Tere is α > 0, uniform wit respect to, suc tat te following olds: α( v V + v S ) sup w V a (v,w ) w V + w S, v V. (39.10) Consequently, te discrete problem (39.6) is well-posed. Proof. Set l := v V + v S and r := sup w V a (v,w ) w V + w S, so tat our goal is to prove tat αl r wit α > 0 uniform wit respect to. (1) Te coercivity of ǎ and te positive semi-definiteness of s imply tat µ 0 v 2 L v 2 M + v 2 S + v 2 S a (v,v ) r l. (39.11) (2) Set w := J (v ), so tat ˇτ 1 2w L = ˇτ 1 2J (v ) L c l owing to (39.8a). Moreover, te boundedness of s from condition (i) implies tat w S c 1 ˇτ 1 2w L. Since y V c 1 τ 1 2y L for all y V (see Exercise 39.1), and since ˇτ τ, we infer from te above bounds tat (3) We now use (39.8b) to infer tat w V + w S (c 1 +c 1)c l. (39.12) c 2 ˇτ 1 2 A1 v 2 L (A 1 v,w ) L +µ 0 v 2 L + v 2 S. (39.13) Te first term on te rigt-and side is rewritten as follows: (A 1 v,w ) L = a (v,w ) (Kv,w ) L 1 2 ((M N)v,w ) L (S v,w ) L s (v,w ). Owing to (39.12), we infer tat a (v,w ) r ( w V + w S ) (c 1 + c 1)c r l. Let δ collect te four remaining terms on te rigt-and side. Using Caucy Scwarz inequalities (in L and for s ), properties (38.25c)- (38.25d) of te boundary fields, and K L (D;K m m ) cµ 0, we infer tat ( ) δ c µ 0 v L w L +( v M + v S ) ρ 1 2 w L + v S w S.

5 Part VIII. First-Order PDEs 529 Using te discrete trace inequality ρ 1 2w L c τ 1 2w L leads to δ c ( µ 0 v 2 L + v 2 M + v 2 S + v 2 S )1 2 ( )1 µ 0 w 2 L + τ 1 2 w 2 L + w 2 S + w 2 2 S cr 1 2 l 3 2, were we used ˇτ τ and te above bounds on ˇτ 1 2w L and w V. Tus, (A 1 v,w ) L c(r l + r 1 2 l 3 2 ). Using once again (39.11) for te last two terms on te rigt-and side of (39.13) leads to ˇτ 1 2A 1 v 2 L c(r l +r 1 2 l 3 2 ). Summing (39.11) to tis last bound yields l 2 c(r l +r 1 2 l 3 2 ). We conclude applying twice Young s inequality. Lemma 39.2 (Boundedness). Tere is c, uniform wit respect to, suc tat ǎ(v,w ) c v V w V olds for all (v,w ) V s V. Proof. Use Lemma Teorem 39.3 (Error estimate). Let u be te unique solution to (38.1) and let u be te unique solution to (39.6) wit stabilization bilinear form s satisfying te design conditions (i)-(ii)-(iii) above. Tere is c, uniform wit respect to, suc tat u u V c inf v V ( u v V + v S ). (39.14) Moreover, if u H r+1 (D;K m ), r [0,k], ten u u V c ( K T (ˇτ 1 K K) 2r+1 K u 2 H r+1 (D K;K m ) were max(β K,µ 0 K ) ˇτ 1 K K max(β D,µ 0 K ). )1 2, (39.15) Proof. Proceeding as in te proof of Strang s First Lemma, we first use stability (Lemma 39.1) to infer tat ǎ(u v,w ) s (v,w ) α( u v V + u v S ) sup, w V w V + w S for all v V. Using boundedness (Lemma 39.2) and te Caucy Scwarz inequality for s leads to u v V u v V + u v S c ( u v V + v S ), wence (39.14) readily follows. Estimate (39.15) ten follows by taking v = I (u) and using te approximation property (39.1) of I (see te proof of Teorem 38.8 for GaLS), togeter wit te design condition (iii) for s and te fact tat τ 1 K ˇτ 1 K for all K T.

6 530 Capter 39. Fluctuation-based stabilization 39.3 Continuous Interior Penalty Te key idea in CIP stabilization (also termed edge stabilization in te literature) is to penalize te jump of A 1 v across mes interfaces. Tis idea as been introduced in Burman [109], Burman and Hansbo [113]; see also [110, 111] for Friedrics systems and te p analysis, and [212] for weigting te edge stabilization in te context of nonlinear conservation laws. For CIP stabilization, we set ˇτ K := min(β 1 D K,µ 1 0 ) for all K T. Recall te discrete averaging operator J g,av introduced in Let φ : D R be te continuous, piecewise affine function suc tat φ(z) = card(t z ) 1 K T z ˇτ K were z is a mes vertex and T z = {K T z K}. Note tat φ is obtained from te piecewise constant function ˇτ by applying te operator J g,av corresponding to te polynomial order k = 1. Te key result underpinning CIP stabilization is te following. Proposition 39.4 (CIP). Assume tat tere is c > 0, uniform wit respect to, suc tat c ˇτ F F [[A 1 v ]] F 2 L(F) µ 0 v 2 L + v 2 S, (39.16) F F for all v V were ˇτ F = max(ˇτ Kl,ˇτ Kr ), for all F F wit F = K l K r, and (A 1 v ) K = d k=1 Ak K k v K, for all K T, wit A k K te meanvalue of A k over K. Ten, (39.8) olds wit J (v ) := J g,av (φa 1 v ). (39.17) a more precise statement will be useful Proof. (1) Let K T. To prove (39.8a), we observe tat te local L 2 -stability of J g,av wen applied to discrete functions (tis follows from Lemma 14.3) implies tat ˇτ 1 2 J (v ) 2 L(K) = ˇτ 1 K g,av J (φa 1 v ) 2 L(K) cˇτ 1 K φa 1v 2 L(D K) c ˇτ 1 2 A1 v 2 L(D K), since mes regularity implies tat φ L (D K) cˇτ K. Summing over K T, we infer tat ˇτ 1 2J (v ) L c ˇτ 1 2A 1 v L. Using a triangle inequality ten leads to ˇτ 1 2 J (v ) L c( ˇτ 1 2 A1 v L + ˇτ 1 2 (A1 A 1 )v L ). Since te fields A k are piecewise Lipscitz wit constant L A cµ 0, using an inverse inequality and te fact tat ˇτ K τ K µ 1 0 leads to ˇτ 1 2(A 1 A 1 )v L cµ v L, wence we infer (39.8a). (2) Concerning (39.8b), we first observe tat φ 1 2 A1 v 2 L = T 1 + (A 1 v,φa 1 v J (v )) L + (A 1 v,φ(a 1 A 1 )v ) L,

7 Part VIII. First-Order PDEs 531 wit T 1 = (A 1 v,j (v )) L. Using Young s inequality leads to 1 2 φ1 2 A1 v 2 L T 1 + φ 1 2 (φa1 v J (v )) 2 L + φ 1 2 (A1 A 1 )v 2 L. Let us denote by T 2,T 3 te two rigtmost terms on te rigt-and side. Owing to Lemma 14.1 applied to te piecewise polynomial φa 1 v, recalling tat FK = {F F F K }, and since φ is a continuous function, we infer tat T 2 = 2 (φa1 v J g,av (φa 1 v )) 2 L(K) K T φ 1 c K T φ 1 L (K) F F K φ 2 L (F) F [[A 1 v ]] F 2 L(F). Te definitions of φ and ˇτ, and mes regularity imply tat ˇτ K cinf x K φ(x) and φ L (K) cˇτ K for all K T (see Exercise 39.3), so tat φ 1 L (K) max φ 2 F FK L (F) cˇτ K. Assumption (39.16) ten implies tat T 2 c(µ 0 v 2 L + v 2 S ). Furtermore, proceeding as above, we infer tat T 3 c ˇτ K µ 2 0 v 2 L(K) cµ 0 v 2 L. K T Collecting tese bounds yields φ 1 2A 1 v 2 L c(t 1+µ 0 v 2 L + v 2 S ), and we conclude tat (39.8b) olds since ˇτ K cinf x K φ(x). Examples of CIP bilinear forms are te following: s CIP (v,w ) = ˇτ F F ([[A 1 v ]] F,[[A 1 w ]] F ) L(F), F F s CIP (v,w ) = F F s CIP (v,w ) = ˇτ F F ([[A 1 v ]] F,[[A 1 w ]] F ) L(F), β F 2 F([[ v ]] F,[[ w ]] F ) L(F), (39.18a) (39.18b) (39.18c) F F wit ˇτ F defined in Proposition 39.4 and β F := max(β Kl,β Kr ) wit F = K l K r. Note tat in (39.18c), only te normal component of te gradient can actually jump across F since functions in V are continuous. Lemma 39.5 (Design conditions). Te bilinear forms s CIP from (39.18) satisfy conditions (i)-(ii)-(iii).

8 532 Capter 39. Fluctuation-based stabilization Proof. (1) Condition (i). Since ˇτ F cmin(ˇτ Kl,ˇτ Kr ) owing to mes regularity and te definition of ˇτ, we infer for (39.18a) tat v 2 S c K T ˇτ K A 1 v 2 L(K) c K T ˇτ K β 2 D 2 K v 2 L(K) c ˇτ 1 2 v 2 L, were we ave used a discrete trace inequality, an inverse inequality, and te fact tat β D 1 K ˇτ 1 K. Tis proves condition (i) for (39.18a), and te proof for (39.18b) and (39.18c) is similar. (2) Condition (ii). It suffices to prove (39.16) owing to Proposition Te proof for (39.18b) is immediate, and tat for (39.18a) ten results from te Lipscitz assumption on te fields A k and an inverse inequality. Finally, concerning (39.18c), te proof follows from ˇτ F F [[A 1 v ]] F 2 L(F) ˇτ F F β 2 F [[ v ]] F 2 L(F) cβ F 2 F [[ v ]] F 2 L(F), since ˇτ F β F 1 F c owing to mes regularity. (3) Finally, condition (iii) follows from te approximation property (39.1). Remark 39.6 (Time-dependent case). Te coice (39.18c) is interesting wit time-dependent fields A k since te matrix associated wit (39.18c) can be assembled only once, wic is not te case for (39.18a)-(39.18b) Two-scale stabilization ( ) In tis section, we present two closely related stabilization tecniques: Local Projection Stabilization (LPS) and Subgrid Viscosity (SGV). Te SGV tecnique as been introduced in Guermond [252, 254] for transport equations, and te LPS tecnique in Braack and Burman [74] for convection-diffusion equations, see also Matties et al. [340]. LPS and SGV bot inge on a twoscale decomposition of te discrete space V, leading to te notions of resolved and fluctuating (or subgrid) scales. Bot stabilization tecniques introduce a least-squares penalty: LPS penalizes te fluctuation of te gradient and SGV penalizes te gradient of te fluctuation. Te notion of scale separation and subgrid scale dissipation is similar in spirit to te spectral viscosity tecnique introduced by Tadmor [436] to approximate nonlinear conservation equations by means of spectral metods. Tis notion is also found in te Ortogonal Subscale Stabilization tecnique of Codina [159] Te two-scale decomposition Te starting point is a two-scale decomposition of V into te form V = R +B. (39.19)

9 Part VIII. First-Order PDEs 533 Note tat te sum is not necessarily direct. Te discrete space R can be viewed as te space of resolved scales, and B as te space of fluctuating (or subgrid) scales. It is important to realize tat, in te present stabilization metods, te degrees of freedom attaced to B only serve to acieve stability, and tat te approximation error is controlled by te best approximation error in te space of resolved scales R (and not in te full space V used in te simulation). We assume te following local approximability property in R : Tereisaquasi-interpolationoperatorI R : V R andaconstantc,uniform wit respect to, suc tat v I R (v) L(K) + (v I R (v)) L(K) c r+1 v H r+1 (D K;K m ), (39.20) for all r [0,k], all v H r+1 (D;K m ), and all K T. Since functions in R are continuous, piecewise polynomials, te components of teir gradients belong to a broken finite element space G = K T G K, were functions in G K are supported in K, i.e., i r G for all r R and all i {1:d}. We assume tat te space of fluctuating scales can also be localized in te form B = K T B K were functions in B K are supported in K (one may tink of functions in B K as bubble-type functions, see te examples below). We define te local L-ortogonal projections πk B : L(K) B K and πk G : L(K) G K for all K T, as well as teir global counterparts π B : L B and π G : L G (wic are assembled cellwise). TekeyassumptionlinkingtelocalgradientspaceG K totelocalfluctuation space B K is te following inf-sup condition introduced in [252] (see also [339]): Tere is γ > 0, uniform wit respect to, suc tat, for all K T, K inf sup gbdx γ, (39.21) g G K b B K g L(K) b L(K) or, equivalently, γ g L(K) π B K g L(K) for all g G K. In wat follows, we consider te local weigting parameter Examples ˇτ K = τ K = min(β K 1 K,µ 1 0 ), K T. (39.22) We can coose for te space of resolved scales R an H 1 -conforming finite element space of degree k 1, say P g k (T ). Ten, G is te broken finite element space P b k 1 (T ); on simplicial affine meses, G K = P k 1,d. Following [252] for k {1,2} and [339] for all k 1, one possibility for coosing B K so tatteinf-supcondition(39.21)oldsistotakeb K = b K G K wereb K iste H 1 0(K)-bubble function proportional to te product of te (d+1) barycentric coordinates over K, see te two upper panels in Figure Instead of working wit bubble functions, one can consider piecewise polynomials for all k 1 [339]. In tis case, te construction starts from te mes defining te space of resolved scales, say Ť. Considering simplicial meses for simplicity, te mes

10 534 Capter 39. Fluctuation-based stabilization Fig Examples of two-scale finite elements. In eac panel, te resolved scales (on te left) are separated from te fluctuating scales (on te rigt). Te resolved scales are eiter P 1 or P 2 Lagrange elements. Te upper panels illustrate te use of a standard bubble function to build te fluctuating scales; te central and lower panels illustrate te use of piecewise polynomial bubble functions on a submes wit te same size (central panel) or alf te size (bottom panel) as tat of te resolved scales space. T defining V is ten built by joining te barycenter of any simplex in Ť to its (d + 1) vertices. Ten we can take V = P g k (T ) and W = P g k (Ť), so tat G = P b k 1 (Ť) and G K = P k 1,d, see te two central panels in Figure Te practical advantage is tat V is a standard Lagrange finite element space. Finally, we mention te two-scale decomposition considered in [252] for k {1,2} wic also offers te advantage of working wit a standard Lagrange finite element space for V, see te two lower panels in Figure Te analysis (not considered erein) is somewat more involved since te fluctuating scales are represented by functions possibly supported on two adjacent mes cells Local Projection Stabilization We define te fluctuation operator κ G = I L π G, were I L is te identity operator in L. Te key idea underpinning LPS stabilization is te following. Proposition 39.7 (LPS). Assume tat tere is c > 0, uniform wit respect to, suc tat c ˇτ 1 2 κ G (A 1 v ) 2 L µ 0 v 2 L + v 2 S, (39.23) for all v V. Ten, (39.8) olds wit Proof. (1) Proving (39.8a) is straigtforward since J (v ) := ˇτπ B π G (A 1 v ). (39.24)

11 Part VIII. First-Order PDEs 535 ˇτ 1 2 J (v ) L = ˇτ 1 2 π B π G (A 1 v ) L ˇτ 1 2 A1 v L, (39.25) owing to te local L-stability of π B and πg. (2) To prove (39.8b), we observe tat ˇτ 1 2 A1 v 2 L = ˇτ 1 2 π G (A 1 v ) 2 L + ˇτ 1 2 κ G A 1 v 2 L ˇτ 1 2 π G (A 1 v ) 2 L +c(µ 0 v 2 L + v 2 S) γ 1 ˇτ 1 2 π B π G (A 1 v ) 2 L +c(µ 0 v 2 L + v 2 S), were we ave used te assumption (39.23) and te inf-sup condition (39.21). Te first term on te rigt-and side, say T 1, can be rewritten as T 1 = (π B π G (A 1 v ),J (v )) L = (π G (A 1 v ),J (v )) L = (A 1 v,j (v )) L (κ G A 1 v,j (v )) L, since J (v ) B. Te last term on te rigt-and side, say T 2, can be bounded using te Caucy Scwarz inequality, Young s inequality, and te bound (39.25), wic implies tat T 2 γ ˇτ 2A 1 1 v 2 L + c γ ˇτ 1 2κ G A 1v 2 L, were γ > 0 can be cosen as small as needed. Te desired bound now follows by using (39.23) to estimate te last term on te rigt-and side. Examples of LPS bilinear forms are te following: s LPS (v,w ) = (ˇτκ G (A 1 v ),κ G (A 1 w )) L, s LPS (v,w ) = (β 2ˇτκ G ( v ),κ G ( w )) L, (39.26a) (39.26b) wit A 1 v defined in Proposition 39.4 and were β : D R is te piecewise constantfunctionsuctatβ K := β K forallk T.WerefertoRemark39.6 for te advantage of considering (39.26b) in time-dependent problems. Lemma 39.8 (Design conditions). Te bilinear forms s LPS from (39.26) satisfy conditions (i)-(ii)-(iii). Proof. Condition (i). Wen s LPS is defined from (39.26a), we use te L- stability of κ G, an inverse inequality, and te fact tat β K 1 K ˇτ 1 K. Te proof is similar wen s LPS is defined from (39.26b). (2) Condition (ii). It suffices to prove (39.23) owing to Proposition Te proof for (39.26a) follows from te triangle inequality ˇτ 2κ 1 G (A 1v ) L ˇτ 1 2κ G (A 1v ) L + ˇτ 1 2κ G ((A 1 A 1 )v ) L, and te last term is bounded using te L-stability of κ G, te Lipscitz property of te fields Ak, and te fact tat L A cµ 0 and ˇτ µ 1 0. Wen slps is defined from (39.26b), we observe tat ˇτ 1 2 κ G (A 1 v ) L(K) d ˇτ 1 2 K A k K κ G ( k v ) L(K) cˇτ 1 2 K β K κ G ( v ) L(K), k=1

12 536 Capter 39. Fluctuation-based stabilization for all K T, were we ave used te linearity of κ G, te triangle inequality, and te fact tat A k K l 2 β K. (3) Finally, condition (iii) is straigtforward to verify since in bot cases for te definition of s LPS, we observe tat IR (u) S = 0. Remark 39.9 (Use of κ G (A 1v )). Defining te LPS bilinear form as s LPS (v,w ) = (ˇτκ G (A 1v ),κ G (A 1w )) L is somewat delicate wen te fields A k are not piecewise constant, since I R(u) S no longer vanises. Bounding tis quantity requires strong regularity assumptions on te fields A k Subgrid Viscosity In te subgrid viscosity metod, te decomposition (39.19) of V is supposed to be a direct sum: V = R B. (39.27) We let π R : V R be te oblique projector based on (39.27), and we define te fluctuation operator κ R = I V π R, wit I V te identity in V. We assume tat (39.27) is L-stable so tat tere is γ R > 0, uniform wit respect to, suc tat γ R π R v L v L, v V. (39.28) Just as for LPS stabilization, we can coose for te space of resolved scales R an H 1 -conforming finite element space of degree k 1, say P g k (T ). Ten, G is te broken finite element space Pk 1 b (T ); on simplicial affine meses, G K = P k 1,d. Te simple coice B K = b K G K is only possible for k d, since oterwise te decomposition (39.27) is no longer direct. For k d+1, a simple possibility is to set B K = b α K G K wit exponent α equal to k+1 d+1 or to te smallest integer larger tan k d+1, see also [252, Prop. 4.1]. Te key idea underpinning SGV stabilization is te following. Proposition (SGV). Assume tat tere is c > 0, uniform wit respect to, suc tat c ˇτ 1 2 A1 (κ R v ) 2 L µ 0 v 2 L + v 2 S, (39.29) for all v V. Ten, (39.8) olds wit J (v ) := ˇτπ B A 1 (π R (v )). (39.30) Proof. (1) Let us start wit (39.8a). We observe tat 1 3 ˇτ 1 2J (v ) 2 L ˇτ 1 2A 1 v 2 L +T 1 +T 2, wit T 1 = ˇτ 1 2 (A1 A 1 )(π R v ) 2 L, T 2 = ˇτ 1 2 A1 (κ R v ) 2 L, were we ave used te triangle inequality and te L-stability of π B. Te Lipscitz property of te fields A k, an inverse inequality, te L-stability of π R from (39.28), and te fact tat L A cµ 0 and ˇτ µ 1 0 imply T 1 cµ 0 v 2 L.

13 Part VIII. First-Order PDEs 537 Te term T 2 is bounded by (39.29). Tis proves (39.8a). (2)Proofof(39.8a).Tetriangleinequalityyields 1 3 ˇτ 1 2A 1 v 2 L T 1+T 2 +T 3 wit T 3 = ˇτ 1 2A 1 (π R v ) 2 L. Since T 1,T 2 ave already been bounded, it remains to estimate T 3. Since A 1 (π R v ) G, we use te inf-sup condition (39.21) and te fact tat J (v ) B to infer tat γ 2 T 3 ˇτ 1 2 π B A 1 (π R v ) 2 L = (π B A 1 (π R v ),J (v )) L = (A 1 (π R v ),J (v )) L = (A 1 (π R v ),J (v )) L ((A 1 A 1 )(π R v ),J (v )) L = (A 1 v,j (v )) L (A 1 (κ R v ),J (v )) L ((A 1 A 1 )(π R v ),J (v )) L, and we denote by T 4,T 5 te last two terms on te rigt-and side. We observe tat T 4 γ ˇτ 1 2 J (v ) 2 L +c γ ˇτ 1 2 A1 (κ R v ) 2 L, owing to te Caucy Scwarz and Young s inequalities, were γ > 0 can be cosen as small as needed. Using te above bound on ˇτ 1 2J (v ) 2 L togeter wit (39.29), we infer tat T 4 γ ˇτ 1 2 A1 v 2 L +c γ (µ 0 v 2 L + v 2 S). For T 5, we proceed similarly and use te above bound on T 1 to infer tat T 5 γ ˇτ 2A 1 1 v 2 L +c γµ 0 v 2 L. Collecting te above bounds leads to ˇτ 1 2 A1 v 2 L γ ˇτ 1 2 A1 v 2 L +c γ ((A 1 v,j (v )) L +µ 0 v 2 L + v 2 S). Taking γ > 0 sufficiently small leads to (39.8b). Examples of SGV bilinear forms are te following: s SGV (v,w ) = (ˇτA 1 (κ R v ),A 1 (κ R w )) L, s SGV (v,w ) = (ˇτA 1 (κ R v ),A 1 (κ R w )) L, s SGV (v,w ) = (β 2ˇτ (κ R v ), (κ R w )) L, (39.31a) (39.31b) (39.31c) were, as above, β : D R is te piecewise constant function suc tat β K := β K for all K T. We also refer to Remark 39.6 for te advantage of considering (39.31c) in time-dependent problems. Lemma (Design conditions). Te bilinear forms s SGV from (39.31) satisfy conditions (i)-(ii)-(iii). Proof. See Exercicse Exercises Exercise 39.1 (Bound on V ). Prove tat y V c τ 1 2y L for all y V, uniformly wit respect to.

14 538 Capter 39. Fluctuation-based stabilization Exercise 39.2 (Inf-sup condition). Verify tat property (39.10) implies tat α v V sup w V a (v,w ) w V for all v V. Exercise 39.3 (Minima). Let a 1,a 2 be two positive real numbers suc tat c 1 a 1 a 2 c 1a 1 for some positive constants c 1,c 1. Let b be a positive real number. Verify tat tere are positive constants c 2,c 2 suc tat c 2 min(a 1,b) min(a 2,b) c 2min(a 1,b). Justify te bounds ˇτ K cinf x K φ(x) and φ L (K) cˇτ K from te proof of Proposition Exercise 39.4 (SGV). Prove Lemma to be cleaned-up Exercise 39.5 (Inf-sup condition). Consider te setting of Prove tat, uniformly wit respect to, ã(r,w ) inf sup = α > 0. r R w V r V w L Exercise 39.6 (Applications). Write possible coices for te CIP, LPS, and SGV stabilization bilinear forms applied to te advection-reaction equation, Darcy s equations, and Maxwell s equations. clean te notation wit stars Exercise 39.7 (Strang s Second Lemma). Te goal of tis exercise is to perform te error analysis using Strang s Second Lemma. Assume tat s can be extended to V V, were V is a subspace of V s and set V := V +V. (i) Prove an inf-sup condition on a using te stability norm v 2 V = µ 0 v 2 L v 2 M + v 2 S + v 2 S + ˇτ 1 2A 1 v 2 L. (ii) Prove tat a (v,w ) c v V w V for all (v,w ) V V wit v 2 V = v 2 V + ˇτ 1 2v 2 L + ρ1 2v 2 L. (iii) Prove tat u u V inf v V u v V + u S. (iv) Apply te above error estimate to CIP and LPS. Can it be applied to SGV?

15 Part VIII. First-Order PDEs 539 Solution to exercises Exercise 39.1 (Bound on V ). We bound te four terms composing y V. For te first term, µ1 20 y L, we use te fact tat µ 0 τ 1. For te second and tird terms, we use te fact tat ρ F cβ KF for all F F (were F = KF D), a discrete trace inequality, and te fact tat β KF 1 K τ 1 F K. Finally, for te fourt term, ˇτ 2A 1 F 1v L, we use te fact tat ˇτ τ, te definition of β K, an inverse inequality, and te fact tat β K 1 K τ 1 K. Exercise 39.2 (Inf-sup condition). Te claim follows from a (v,w ) a (v,w ) α v V α( v V + v S) sup sup. w V w V + w S w V w V Exercise 39.3 (Minima). We distinguis four cases. (i) min(a 1,b) = min(a 2,b); ten, te desired bound olds wit c 2 = c 2 = 1. (ii) min(a 1,b) = b and min(a 2,b) = a 2; ten, te desired bound olds wit c 2 = c 1 and c 2 = 1. (iii) min(a 1,b) = a 1 and min(a 2,b) = b; ten, te desired bound olds wit c 2 = 1 and c 2 = c 1. (iv) min(a 1,b) = a 1 and min(a 2,b) = a 2; ten, te desired bound olds wit c 2 = c 1 and c 2 = c 1. Consider now te bounds ˇτ K cinf x K φ(x) and φ L (K) cˇτ K. Since φ is affine in K, tere is a vertex a K suc tat inf x K φ(x) = φ(a). Owing to te definition of φ and te fact tat ˇτ takes positive values, we infer tat φ(a) card(t a) 1ˇτ K. Tis proves te first bound. For te second bound, we observe tat te above reasoning on te minima combined wit mes regularity and te definition of ˇτ imply tat max L TK ˇτ L cˇτ K, were T K = {K T K K }; ence, since φ L (K) = φ(a ) for some vertex a K, te definition of φ implies tat φ L (K) max L TK ˇτ L cˇτ K. Tis concludes te proof. Exercise 39.4 (SGV). Exercise 39.5 (Inf-sup condition). Exercise 39.6 (Applications). For te advection-reaction equation, using (39.18a) leads to s CIP (v,w ) = ρ 1 β 2 F ([β v ] F, [β w ] F) L 2 (F), F F wit ρ β = β L (D). Alternatively, using (39.18c) leads to s CIP (v,w ) = F F ρ β 2 F (nf [ v ] F,n F [ w ] F) L 2 (F). Exercise 39.7 (Strang s Second Lemma). (i) (ii) (iii) (iv) For CIP, we need first to extend s CIP to te space V = [H t (D)] m wit t > 3 2 ; ten, observing tat u S = 0, we obtain u u V ρ 1/2 A k+1/2 u [H k+1 (D)] m. For LPS, te same bound is obtained for te two coices (39.26a) and (39.26b) of s LPS since u S = u I R (u) S ρ 1 2A ˇτ 1 2 (u I R (u)) L, wic is bounded by te rigt-and side of (??). Finally, Teorem?? is not suitable for SGV, since te stabilization bilinear form s SGV is meaningful only for discrete arguments.