LOCAL A POSTERIORI ERROR ESTIMATES FOR TIME-DEPENDENT HAMILTON-JACOBI EQUATIONS

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1 MATHEMATICS OF COMPUTATION Volume 82, Number 28, January 23, Pages S (22)26-X Aricle elecronically published on June 5, 22 LOCAL A POSTERIORI ERROR ESTIMATES FOR TIME-DEPENDENT HAMILTON-JACOBI EQUATIONS BERNARDO COCKBURN, IVAN MEREV, AND JIANLIANG QIAN Absrac. In his paper, we obain he firs local a poseriori error esimae for ime-dependen Hamilon-Jacobi equaions. Given an arbirary domain Ω and a ime T, he esimae gives an upper bound for he L -norm in Ω a ime T of he difference beween he viscosiy soluion u and any coninuous funcion v in erms of he iniial error in he domain of dependence and in erms of he (shifed) residual of v in he union of all he cones of dependence wih verices in Ω. The esimae holds for general Hamilonians and any space dimension. I is hus an ideal ool for devising adapive algorihms wih rigorous error conrol for ime-dependen Hamilon-Jacobi equaions. This resul is an eension o he global a poseriori error esimae obained by S. Alber, B. Cockburn, D. French, and T. Peerson in A poseriori error esimaes for general numerical mehods for Hamilon-Jacobi equaions. Par II: The ime-dependen case, Finie Volumes for Comple Applicaions, vol. III, June 22, pp Numerical eperimens invesigaing he sharpness of he a poseriori error esimaes are given.. Inroducion In his paper, we obain he firs local a poseriori error esimae beween he viscosiy soluion u of he following Cauchy problem for he model Hamilon-Jacobi equaion, (.) u + H(, u) = for(, ) R d (,T), (.2) u(, =)=u () for R d, and any coninuous funcion v. The esimae has he form (.3) u(t ) v(t ) L (Ω) Φ(v; T,Ω), where T is an arbirary posiive number, Ω is an arbirary sub-domain of R d,andas epeced, he funcional Φ depends on he error in he domain of dependence a he iniial ime and on he (shifed) residual of he funcion v on he union of all cones of dependence wih verices in Ω. The resuls in his paper eend hose obained in [2] and [3]. Indeed, when Ω = R d, our esimae becomes he global esimae obained in [3], which was in urn obained by eending o he ime-dependen case he esimae for he seady-sae case proposed in [2]. Received by he edior May 7, 2 and, in revised form, May 27, 2. 2 Mahemaics Subjec Classificaion. Primary 65M5, 65M2; Secondary 49L25. Key words and phrases. A poseriori error esimaes, Hamilon-Jacobi equaions. The firs auhor was parially suppored by he Naional Science Foundaion (Gran DMS ) and by he Minnesoa Supercompuing Insiue. The hird auhor was parially suppored by he Naional Science Foundaion (NSF 84 and NSF 836). 87 c 22 American Mahemaical Sociey

2 88 B. COCKBURN, I. MEREV, AND J. QIAN Several auhors have obained error esimaes in he L -norm for Hamilon- Jacobi equaions, bu he resuls obained in [2] and [3], and, hence, he resul in his paper, are differen in many aspecs. Crandall and Lions [9] obained an a priori esimae for he difference beween he viscosiy soluion u and he approimaion v given by a monoone scheme defined on Caresian grids. In our seing heir resul canbewrieninheform (.4) u(t ) v(t ) L (Ω) C Δ, where Δ is he maimum mesh size and C> is a consan. Souganidis [5] eended his esimae o more general Hamilonians and o general finie difference schemes on Caresian grids. Abgrall [] inroduced he inrinsic monoone schemes for unsrucured meshes and proved ha he same error esimae holds. Falcone and Ferrei [] obained a priori error esimaes for Hamilon-Jacobi-Bellman equaions for schemes ha were consruced by using he discree dynamic programming principle and were devised o converge very fas o he eac soluion. They assume ha he viscosiy soluion is very smooh, however, and heir resuls apply only o conve Hamilonians. Qian [4] proved ha (.4) holds for a modificaion of he classic La-Friedrichs scheme ha allows for locally varying ime and space grids. All hese a priori error esimaes depend on he numerical mehod used o compue he approimae soluion v, involve informaion ha is no known abou he eac soluion, and canno capure he feaures of he paricular problem under consideraion. In conras, he a poseriori error esimae (.3), as well as he a poseriori error esimaes obained in [2] and [3], hold regardless of how v is compued. They can hus be applied if v is obained by means of a finie difference scheme such as he ENO scheme developed by Osher and Shu [3], a finie elemen mehod such as he disconinuous Galerkin (DG) mehod proposed by Hu and Shu [], he Perov-Galerkin mehod used by Barh and Sehian [4], or a finie volume mehod such as he inrinsic monoone scheme of Abgrall []. These esimaes are also independen of he smoohness of he eac soluion and ake ino accoun he pariculariies of he specific problem. This makes hem an ideal ool for devising adapive algorihms wih rigorous error conrol for Hamilon-Jacobi equaions. The seady-sae a poseriori error esimae obained in [2] has been uilized in a few adapive schemes. An adapive mehod for one- and wo-dimensional seady sae Hamilon-Jacobi equaions using a monoone scheme was developed in [6] and [7]. An adapive scheme for seady-sae Hamilon-Jacobi equaions using he disconinuous Galerkin (DG) mehod was proposed in [5]. An adapive mehod for he ime-dependen case is a subjec ha will be eplored in a forhcoming paper. The paper is organized as follows. We sae and prove he local a poseriori error esimae in secion 2. In his secion we follow [3] closely. We devoe secion 3 o he numerical sudy of he efficiency of he a poseriori error esimae when he approimae soluion v is compued using a monoone scheme. Finally, we end in secion 4 wih some concluding remarks. 2. A poseriori error esimaes 2.. The viscosiy soluion. We begin by recalling he definiion of he viscosiy soluion. To sae he definiion, we need he noion of semi-differenials of a funcion defined on R d (,T). The superdifferenial D + u(, ) of a funcion u a

3 ERROR ESTIMATES FOR HAMILTON-JACOBI EQUATIONS 89 apoin(, ) R d (,T)isheseofallvecorsp =(p,p )inr d R such ha ( ) u(y, s) {u(, )+(s ) p +(y ) p } lim sup, (y,s) R d (,T ) (,) (y, s) (, ) and he subdifferenial D u(, ) ofu a a poin (, ) R d (,T)isheseofall vecors p =(p,p )inr d R such ha ( ) u(y, s) {u(, )+(s ) p +(y ) p } lim inf. (y,s) R d (,T ) (,) (y, s) (, ) The definiion of he viscosiy soluion of (.), (.2) is saed in erms of he residual, which is defined as R(, p) =p + H(, p ). Definiion 2. ([8]). A viscosiy soluion u of he iniial-value problem for he Hamilon-Jacobi equaion (.), (.2) is a coninuous funcion on R d (,T) saisfying u(, =)=u () such ha, for all (, ) inr d (,T), σr(, p), p D σ u(, ), σ {+, } The a poseriori error esimae. Given any sub-domain Ω of R d and any ime T >, he local a poseriori error esimae ha we propose gives an upper bound for he semi-norms u v,ω,t = sup (u(, T ) v(, T )) +, Ω u v +,Ω,T = sup (v(, T ) u(, T )) +, Ω in erms of he behavior of he funcion v on he se Q T = Ω V (T ) {}, (,T ) where Ω V = { : dis(, Ω) V}, and in erms of he quaniy u v σ,ωv T,. The posiive parameer V is defined as (2.5) V =sup H(, ) Lip, R d where he righ-hand side is assumed o be finie. This ensures ha if a characerisic his he border of Q T,idoesnoenerinoQ T.Moreover,Q T is nohing bu (he inersecion wih R d (,T) of) he union of he cones of dependence wih verices in Ω; see Figure below. In order o give he saemen and he proof of he error esimae, we need o inroduce he following noaion. For any vecor p =(p,p ) R d R and ɛ =(ɛ,ɛ ) wih ɛ,ɛ >, he shifed residual R ɛ is defined as (2.6) R ɛ (, p) =p + H( ɛ p,p ). For κ =(κ,κ ) R 2, he paraboloid P v is given by (2.7) P v (,, p, κ; y, s) =v(, )+(y, s ) p + κ 2 s 2 + κ 2 y 2,

4 9 B. COCKBURN, I. MEREV, AND J. QIAN T Ω {T} Ω Ω {} VT Figure. The region enclosed by he rapezoid Q T. The viscosiy soluion u in Ω {T } does no depend on is iniial values ouside of Ω VT. for all (y, s) R d [,T]. We also need he quaniy (2.8) ω ɛ (v;,, p) =v(, ) v( ɛ p, ) ɛ 2 p ɛ The quaniy ω ɛ incorporaes ino he esimae informaion abou he smoohness of he funcion v. I can be bounded in erms of he moduli of coninuiy of v. For eample, if v( =) W, (R d )andv L (,T; R d ), (2.9) ω σɛ (v;,, p) v 2 L (,T ;R d ) 2 ɛ + v() 2 W, (R d ) ɛ. 2 Wih he noaion inroduced above, we have he following resul. Theorem 2.2. Le u be he viscosiy soluion of he problem (.), (.2) and le v be any coninuous funcion on R d [,T]. LeΩ be a sub-domain of R d.then,for σ {, +}, we have ha (2.) u v σ,ω,t u v σ,ωv T, + inf Φ σ(v; ɛ), ɛ,ɛ > where { (σωσɛ (v;,, p) ) + + ( σtrσɛ (, p) ) + }. (2.) Φ σ (v; ɛ) = sup ((,),p) A σ (v;ɛ) The se A σ (v; ɛ) is he se of poins ((, ),p) in Q T R d+ such ha (2.2) σ {v(y, s) P v (,, p, (σ/ɛ,σ/ɛ ); y, s)} (y, s) Q T. Remark. As saed in he inroducion, he above esimae reduces o he global a poseriori error esimae obained in [3] when Ω = R d.

5 ERROR ESTIMATES FOR HAMILTON-JACOBI EQUATIONS 9 Remark 2. Noe ha since ma h σ,ω,t = h(, T ) L (Ω), σ={,+} from he esimaes of he semi-norms u v σ,ω,t,weobain u(, T ) v(, T ) L (Ω) u(, ) v(, ) L (Ω VT ) + ma σ {,+} inf Φ σ(v; ɛ). ɛ,ɛ > When v saisfies cerain smoohness condiions, such as he condiions required for (2.9) o hold or v is Lipschiz in boh variables, we can le ɛ and ɛ go o zero in (2.); using (2.9), we obain u v σ,ω,t u v σ,ωv T, + sup (σtr(, p)) +, ((,),p) Q T D σ v(,) which furhermore implies ha u(, T ) v(, T ) L (Ω) u(, ) v(, ) L (Ω VT ) { + ma σ {,+} } sup (σtr(, p)) +. ((,),p) Q T D σ v(,) This esimae is sharp when he viscosiy soluion u is smooh and v has a nonoscillaory residual, as will be seen from he numerical eperimens. However, for a nonsmooh viscosiy soluion u, he righ-hand side of he above inequaliy migh remain of order one while v converges o he viscosiy soluion [2]. Remark 3. The poin ((, ),p) belongs o he se A (v; ɛ) if he graph of v remains above he paraboloid P v (,, p, ( /ɛ, /ɛ );, ). Since he paraboloid ouches v a (, ), when (, ) is in he inerior of Q T, p belongs o D v(, ). This so-called paraboloid es, i.e., esing wheher ((, ),p) belongs o A σ (v; ɛ), can be used o deec he locaion of he disconinuiies in he gradien of he eac soluion u a any ime [,T]. Consider a viscosiy soluion u ha has a kink a he poin (, ), for eample. Le v be a smooh funcion ha is, roughly speaking, very close o he viscosiy soluion u. For a given value of he auiliary parameer ɛ =(ɛ,ɛ ), here is an inerval around (, ) such ha he poins inside his inerval fail he paraboloid es. We illusrae his fac when we discuss he resuls of our numerical eperimens for nonsmooh viscosiy soluions. Remark 4. We also wan o sress he role played by he auiliary parameer ɛ = (ɛ,ɛ ). The se A σ (v; ɛ) decreases as each of he componens of he auiliary parameer ɛ increases. This induces a endency on Φ σ (v; ɛ) o decrease. A he same ime, he shifed residual σr σɛ (, p) migh increase. The opimal value of ɛ is obained by balancing hese wo endencies. Remark 5. Le us argue why i is reasonable o epec ha our a poseriori error esimae holds when he rapezoidal domain Q T is replaced by a smaller se we denoe by Γ Ω,T. Le us moivae he definiion of Γ Ω,T. From he heory of characerisics we know ha he region enclosed by he rapezoid Q T is larger han he smalles compac domain ha conains he characerisics of he iniial-value problem (.), (.2) ha ener ino Ω a ime T or prior o ime T have enered ino a shock ha is inside Ω a ime T. We denoe his compac domain by Γ Ω,T.Iisevidenfrom equaions (2.) and (2.2) ha he larger he region enclosed by he rapezoid Q T, he larger he error esimae. This is especially rue in he case when he

6 92 B. COCKBURN, I. MEREV, AND J. QIAN viscosiy soluion is smooh inside Ω, bu he rapezoid Q T conains a par of he pah of he kink in he viscosiy soluion, as would be illusraed by he numerical eperimens. Since he viscosiy soluion in Ω a ime T is compleely deermined by he iniial condiion in he domain of dependence of Ω and is behavior along he characerisics ha ener ino Ω a ime T, i is reasonable o epec ha a sharper error esimae would be obained if we replace he rapezoid Q T by a smaller compac domain ha beer approimaes Γ Ω,T. We could deermine he se Γ Ω,T by solving a Hamilon-Jacobi iniial-value problem. The idea is as follows. Assume ha he Hamilonian funcion H(, p) is differeniable wih respec o p, and assume ha he viscosiy soluion u of he iniial-value problem (.), (.2) has a gradien a all poins (, ) R d (,T). Now, le ϕ(, ) be he soluion o he erminal-value problem (2.3) ϕ + p H(, u) ϕ = for(, ) R d (,T), { for Ω, ϕ(, T )= for / Ω. I is easily seen ha he characerisics of (2.3) are also characerisics of (.) for [,T]. The soluion o (2.3) is equal o along he characerisics ha are ouside Ω a ime T and is equal o along he characerisics inside he domain Γ Ω,T. Since u is no known a priori, o approimae he se Γ Ω,T, we could replace u by an approimaion v obained using a numerical scheme such as he La-Friedrichs scheme [9]. Then, we would solve (2.3) numerically o obain an approimae soluion ϕ h and approimae Γ Ω,T by he domain where ϕ h is nonzero. As we are going o see, our numerical eperimens do confirm ha his approach works very well. The heoreical jusificaion of he algorihm, however, is sill an open problem Proof of he a poseriori error esimae. The proof we presen ne is a modificaion of he one presened in [3] for he case Ω = R d. We only prove he resul for σ = since he proof for σ = + is enirely analogous. We proceed in several seps. Sep (The auiliary quaniy Δ δ ). Noe ha Definiion 2. of he viscosiy soluion holds only in he inerior of Q T. So insead of working wih he quaniy Δ= sup (u(, ) v(, )) u v,ωv T,, (,) Q T we consider he auiliary quaniy Δ δ = sup (,) Q T ( u(, ) v(, ) where δ is an arbirary posiive parameer, Ψ(, ) =T φ(), ) δ u v,ωv Ψ(, ) T,, and he funcion φ is consruced in such a way ha Ψ(, ) > in he inerior of Q T and ha Ψ(, ) =for(, ) inhese Q T \ Ω VT {}. I is no difficul o see ha he funcion φ is nohing bu he (suiably scaled) disance funcion o he se Ω. More precisely, φ is idenically equal o zero in Ω,

7 ERROR ESTIMATES FOR HAMILTON-JACOBI EQUATIONS 93 and ouside Ω, i is he viscosiy soluion of he eikonal equaion V φ() = for Ω c, φ() = for Ω. Clearly, for [,T], we have (2.4) Ω V (T ) = { R d Ψ(, ) > }. Sep 2 (The auiliary quaniy Δ δ,ν ). In order o avoid some echnical difficulies originaing from he fac ha φ may no be differeniable, we are going o replace i by a smooher funcion. To do so, we choose a smooh, nonnegaive funcion η wih inegral equal o one and suppor in he uni ball and consider he funcion φ ν = η ν φ, where η ν () = η( ν d ν ) for all ν>, and denoes he convoluion wih respec o. Clearly, φ ν is a smooh funcion. Accordingly, we replace Ψ(, ) byψ ν (, ) = T φ ν (). Now, insead of considering he quaniy Δ δ,weake ( ) Δ δ,ν = u(, ) v(, ) u v,ωv T,, sup (,) Q T,ν δ Ψ ν (, ) where (2.5) Q T,ν = {(, ) R d (,T) Ψ ν (, ) > }. Noe ha for every (, ) Q T we can wrie δ (2.6) u(, ) v(, ) u v,ωv T, +Δ δ,ν + Ψ ν (, ), for some ν<ν, ν >, because of he definiions of Q T and Q T,ν, (2.4) and (2.5), and he fac ha lim ν φ ν () =φ() for all R d since φ is coninuous. The idea of he proof is o obain an esimae of he form Δ δ,ν C δ δ, for some consans C andδ >. If we hen le δ in (2.6), we obain he desired resul, namely, (2.7) u v,ω,t u v,ωv T, + C. Sep 3 (The auiliary funcion ψ). Assume now ha for some δ>andν>, Δ δ,ν > ; oherwise, here is nohing o prove. We inroduce he auiliary funcion ψ(, ; y, s) defined by ψ(, ; y, s) =u(, ) v(y, s) δ + s) ( θ)( Ψ ν (, ) 2 T Δ δ,ν ( s)2 y 2, 2ɛ 2ɛ where θ (, ) is a parameer o be deermined laer. Since boh u and v are coninuous and Q T,ν is bounded, here eiss a poin (ˆ, ˆ;ŷ, ŝ) Q T,ν Q T,ν such ha (2.8) ψ(ˆ, ˆ;ŷ, ŝ) ψ(, ; y, s) (, ; y, s) Q T,ν Q T,ν.

8 Using he fac ha ( Ψν (ˆ, ˆ), Ψ ν (ˆ, ˆ) ) =( φ ν (ˆ), ), 94 B. COCKBURN, I. MEREV, AND J. QIAN Noe ha by definiion of Q T,ν, (2.5), Ψ ν (, ) for all he poins (, ) Q T,ν. Since Ψ ν (ˆ, ˆ) canno be equal o zero, we mus have ha Ψ ν (ˆ, ˆ) >. This implies ha eiher (ˆ, ˆ) belongs o he domain Q T,ν or ˆ =. Moreover, if we se ( (ˆ ŷ) ( θ) ˆp =(ˆp y, ˆp s )=, ɛ 2 T Δ δ,ν + (ˆ ) ŝ), ɛ he poin ((ŷ, ŝ), ˆp) belongs o he se A (v; ɛ). Indeed, aking = ˆ and =ˆ in (2.8), we obain v(y, s) v(ŷ, ŝ)+ˆp (y ŷ, s ŝ) s ŝ 2 y ŷ 2 2ɛ 2ɛ = P v (ŷ, ŝ, ˆp, ( /ɛ, /ɛ ); y, s). Sep 4(Thecaseˆ >). Assume ha ˆ >. In his case, since he mapping (, ) ψ(, ;ŷ, ŝ) aainsamaimuma(ˆ, ˆ) Q T,ν,wehaveha ( ) D,ψ(ˆ, + ˆ;ŷ, ŝ) =D + ( θ) u(ˆ, ˆ), Δ δ,ν T δ ( + Ψν (ˆ, ˆ), Ψ 2 Ψ ν (ˆ, ˆ) ) ˆp. ν(ˆ, ˆ) Hence, δ ( Ψν (ˆ, ˆ), Ψ 2 Ψ ν (ˆ, ˆ) ) ( ) ( θ) +, Δ δ,ν +ˆp D ν(ˆ, ˆ) + u(ˆ, ˆ). T Since u is he viscosiy soluion, Definiion 2. implies ha ( ) δ Ψ 2 ν(ˆ, ˆ) ( θ) δ Ψ ν (ˆ, ˆ)+ Δ δ,ν +ˆp s + H ˆ, ˆp y T Ψ 2 ν(ˆ, ˆ) Ψ ν(ˆ, ˆ), which can be wrien as ( θ) (2.9) Δ δ,ν ˆp s H(ˆ, ˆp y )+Θ, T where ( ) δ Θ= Ψ 2 ν(ˆ, ˆ) δ Ψ ν (ˆ, ˆ)+H(ˆ, ˆp y ) H ˆ, ˆp y Ψ 2 ν(ˆ, ˆ) Ψ ν(ˆ, ˆ). Ne, we show ha Θ. Indeed, by he definiion of he parameer V,(2.5), Θ δ Ψ 2 ν(ˆ, ˆ) δ Ψ ν (ˆ, ˆ)+V Ψ 2 ν(ˆ, ˆ) Ψ ν(ˆ, ˆ). we obain δ Θ Ψ 2 ν(ˆ, ˆ) ( V φ ν(ˆ) ). Since (2.2) V φ ν () R d, we conclude ha Θ. To see ha he above inequaliy holds, i suffices o realize ha he definiion of φ implies ha φ = in Ω, and ha ouside Ω, φ is

9 ERROR ESTIMATES FOR HAMILTON-JACOBI EQUATIONS 95 /V imes he disance-o-ω funcion. This implies ha all he elemens p of is superdifferenial and subdifferenial yield V p. Inequaliy (2.2) is a reflecion of his simple fac. Since Δ δ,ν > andˆ =ŷ + ɛ ˆp y, using (2.9) we obain which implies ha Δ δ,ν T θ ( ˆp s H(ŷ + ɛ ˆp y, ˆp y )) +, (2.2) Δ δ,ν T θ ( R ɛ(ŷ, ˆp)) +. Sep 5(Thecaseˆ =). Now, le us consider he case in which ˆ =. Δ δ,ν >, θ>and /T <, sup (,) Q T,ν ψ(, ;, ) sup (,) Q T,ν ( u(, ) v(, ) ) δ ( θ)δ δ,ν Ψ ν (, ) Since =Δ δ,ν + u v,ωv T, ( θ)δ δ,ν = θ Δ δ,ν + u v,ωv T,. Therefore, θ Δ δ,ν + u v,ωv T, ψ(ˆ, ; ŷ, ŝ) u(ˆ, ) v(ŷ, ŝ) ŝ2 ˆ ŷ 2 2ɛ 2ɛ u v,ω, + v(ˆ, ) v(ŷ, ŝ) ŝ2 ˆ ŷ 2, 2ɛ 2ɛ and finally, since Δ δ,ν >, ˆ =ŷ + ɛ ˆp y and u v,ω, u v,ωv T,, we obain (2.22) Δ δ,ν θ ( ω ɛ(v;ŷ, ŝ, ˆp)) +. Sep 6 (Conclusion). Puing ogeher he bounds (2.2) and (2.22) yields { } A Δ δ,ν ma θ, B, θ where A =( ω ɛ (v;ŷ, ŝ, ˆp)) + and B = T ( R ɛ (ŷ, ˆp)) +. Now, we easily see ha Δ δ,ν A + B, by aking he limi as θ ends o A/(A + B) [, ]. This implies ha (2.7) holds wih C = A + B and Theorem 2.2 follows. 3. Numerical eperimens The purpose of his secion is o sudy he applicaion of he a poseriori error esimae o approimaions generaed hrough numerical schemes; we consider a sandard monoone scheme. Since he error esimae gives only an upper bound for he L -norm of he difference beween he viscosiy soluion u and any approimaion v, we are going o numerically sudy is sharpness on several prooypical

10 96 B. COCKBURN, I. MEREV, AND J. QIAN one-dimensional problems. In oher words, we are going o sudy how close he so-called effeciviy inde is o, which is defined as Φ(v; T,Ω) (3.23) ei(u, v) =, u(t ) v(t ) L (Ω) where Ω is an arbirary sub-domain of R d. In he case when Ω = R d or Ω = Ω per, where Ω per is a sub-domain of R d ha is equal in size o he period of he viscosiy soluion u, we obain he effeciviy inde for he global a poseriori error esimae. To ease he noaion we use D o denoe he domain over which we esimae he a poseriori error esimaes, i.e., D = Q T in he case of he local a poseriori error esimae wih he rapezoid as he domain, D = Γ Ω,T in he case of he local esimae wih Q T replaced by Γ Ω,T as eplained in he previous secion, and D = R d (,T) in he case of he global a poseriori error esimae. For he sake of simpliciy, we use uniform grids in all our eperimens. 3.. Discreizaion of he norms and he nonlinear funcionals of he a poseriori error esimaes. Since he approimae soluions are defined by a finie number of degrees of freedom, we replace he domain D over which we evaluae he funcionals Φ σ ( ;, ) by a finie number of poins inside D which we denoe by D h. We also replace he domain Ω {T } over which we evaluae he L -norm of he difference beween he viscosiy soluion u and is approimaion v by a finie seofpoinsinω {T } ha we denoe by Ω h,t. Theoreically, he evaluaion of he error esimaes requires opimizaion over he se ɛ,ɛ [, ). In pracice, however, i is sensible o replace [, ) wihhe se (3.24) E h = {(ɛ,ɛ )=i (ω /2,ω /2), i 4 ln (/ω ) }, where ω, given by (3.3), is an upper bound for he arificial diffusion coefficien of he numerical scheme under consideraion, and ω is given by ω = ω Δ Δ. We denoe by Φ h,σ ( ) he funcionals in he a poseriori error esimaes obained afer he above menioned modificaions. To simplify he noaion, we also le Φ h (v) = ma Φ h,σ(v). σ {,+} The effeciviy inde ei(u, v) is accordingly replaced by he compuaional effeciviy inde (u, v) =Φ h (v)/ u v L (Ω h,t ) Evaluaion of he shifed residual on Q T. Noe ha if (, ) is a poin on Q T, he paraboloid es could be saisfied by vecors p =(p,p ) R d R ha are no necessarily in he semi-differenials of v a (, ), making he evaluaion of he shifed residual a nonrivial maer. Consider he case when σ =. Recall ha he se A (v; ɛ) isheseofpoins((, ),p) Q T R d+ such ha v(y, s) v(, )+p (y, s ) s 2 y 2 (y, s) Q T, 2ɛ 2ɛ which implies ha v(y, s) v(, ) p (y, s ) (3.25) lim inf (y, s) Q T. (y,s) (,) (y, s) (, )

11 ERROR ESTIMATES FOR HAMILTON-JACOBI EQUATIONS 97 Le (y, s) (, ) ν = (y, s) (, ). For simpliciy assume ha v is coninuously differeniable in a neighborhood of he poin (, ), as is he case in many pracical applicaions. In his case he wo semi-differenials reduce o he eac differenial of v and (3.25) implies ha (3.26) (Dv(, ) p) ν, which we can wrie as δ ν by leing δ =(Dv(, ) p). In paricular, (3.26) holds for all ν ha are inward normal o he boundary of he rapezoid Q T.Ne, we show ha in he case when H is a funcion of p only,asishecaseinalloures problems, o compue he a poseriori error esimae, he shifed residual needs o be evaluaed no a all poins ((, ),p) ha pass he paraboloid es, bu only a he poins ((, ),Dv(, )). Indeed, assuming ha H is differeniable in and p, in (2.) he erm ( R ɛ (, p)) + saisfies ( R ɛ (, p)) + =( [p + H( + ɛ p,p )]) + where =( [v (, )+H( + ɛ v(, ), v(, ))] + Υ) + ( R ɛ (, v(, ))) + +Υ +, Υ= δ H ( + ɛ ( v(, )+δ ), v(, )+δ )+H ( + ɛ v(, ), v(, )) d = δ dη H ( + ɛ ( v(, )+ηδ ), v(, )+ηδ ) dη = δ δ ɛ H ( + ɛ ( v(, )+ηδ ), v(, )+ηδ ) dη δ p H ( + ɛ ( v(, )+ηδ ), v(, )+ηδ ) dη. In he case when H is a differeniable funcion of p only, Υ reduces o ( ) Υ= δ, p H ( + ɛ ( v(, )+ηδ ), v(, )+ηδ ) dη, where in he second inequaliy we have used (3.26) and he definiion of he boundary of Q T. Therefore, ( R ɛ (, p)) + is bounded above by ( R ɛ (, v(, ))) Fas evaluaion of he paraboloid es. To evaluae he paraboloid es (2.2) is compuaionally very epensive. To deermine if he poin ((, ),p) D R d+ belongs o he se A σ (v; ɛ), we mus compare P v (,, p, (σ/ɛ,σ/ɛ ); y, s) and v(y, s) foreachpoin(y, s) D. When v is Lipschiz, however, which is he case in mos pracical applicaions, i is no necessary o perform his comparison for all (y, s) in he domain D, bu only on a significanly smaller se. Noe ha if P v is angen o v a he poin (y, s), we mus have ha ( q =(q,q )= p + σ (y ),p + σ ) (s ) ɛ ɛ

12 98 B. COCKBURN, I. MEREV, AND J. QIAN for some q D σ v(y, s). This implies ha (3.27) y = p q ɛ 2ɛ v(,s) Lip(D,s ) and (3.28) s = p q ɛ 2ɛ v(y, ) Lip(Dy, ), where D,s = { y R d (y, s) D } and D y, = {s (,T) (y, s) D}. Thus, we can replace (2.2) by he following condiion: σ {v(y, s) P v (,, p, (σ/ɛ,σ/ɛ ); y, s)} (y, s) D : (3.27), (3.28) hold. In our compuaions, we acually use a discree version of he above paraboloid es, which is carried ou only for (, ), (y, s) D h Tes problems. To sudy he efficiency of he global and local a poseriori error esimaes, we consider he model problem (3.29) u + H (u )= for(, ) [, ] (, ), u(,)=u(,) for [, ), u(, ) = u () for [, ], wih differen Hamilonian funcions and iniial condiions. Our es problems are chosen in order o cover he main cases of possible behavior of he viscosiy soluion. Thus, hey display smooh and nonsmooh soluions for linear, conve, and nonconve Hamilonians; similar choices have been aken, for eample, in [2, 6, 7, 4, 5]. Wih he noaion ha we inroduced above, Ω per =[, ]. Table summarizes he Hamilonian funcions H(p), he iniial condiions u (), he ime inervals of consideraion, and he smoohness of he viscosiy soluions u a ime T. Table. Tes problems. Problem H(p) u () [,T] u(, T ) (L).75p sin(π) [,.5] smooh { +sin(.5π), [, ] (L2).75p [,.5] nonsmooh.5+.5cos(π), (, ] (C).5(p +) 2 cos(π)/π (NC) cos(p +) cos(π) [,.2] smooh [,.4] nonsmooh [,.5/π 2 ] [,.5/π 2 ] smooh nonsmooh Noe ha problems (L) and (L2), for which he Hamilonian is a linear funcion, are eamples of one-dimensional ranspor equaions wih consan coefficiens, and heir eac soluions are given by u(, ) =u (.75). Furhermore, noe ha he spaial derivaive of he viscosiy soluion of a firs order one-dimensional Hamilon-Jacobi equaion solves a corresponding conservaion law. We can use he heory of characerisics and shock fiing [6] o obain informaion abou he shocks of he corresponding conservaion laws. In his way we can find he locaion of he kinks in he viscosiy soluions of our es problems a any ime. Figure 2 shows he eac soluions of he es problems a he end of he ime inervals considered.

13 ERROR ESTIMATES FOR HAMILTON-JACOBI EQUATIONS u(,t) u(,t) (a) Problem (L); T = (b) Problem (L2); T =.5. u(,t).2.2 u(,t) (c) Problem (C); T = (d) Problem (C); T = u(,t) u(,t) (e) Problem (NC); T =.5/π (f) Problem (NC); T =.5/π 2. Figure 2. Eac soluions of he es problems The monoone scheme. In our numerical eperimens we use a monoone scheme on a uniform grid. Given he mesh sizes Δ, Δ>, he value of our numerical approimaion o he viscosiy soluion u of (3.29) a ( j, n )=(jδ, nδ), j, n Z, will be denoed by v n j.weusehefollowingscheme (3.3) v n+ j [ = vj n Δ H ( v n j+ v n j 2Δ ) θ λ ( v n j+ 2v n j + vn j Δ )], where θ> is given and λ =Δ/Δ. This scheme is monoone as long as 2θ (monooniciy in v n j ), and θ λ H (α) /2 forα R (monooniciy in v n j+ and

14 2 B. COCKBURN, I. MEREV, AND J. QIAN vj n ) [9]. In equaion (3.3), we ake θ = λ sup α R H (α), 2 and fi λ in such a way ha 2θ holds. This choice of θ yields he well-known La-Friedrichs scheme. The use of he scheme (3.3) is jusified by he fac ha i is guaraneed o converge o he correc viscosiy soluion [9]. Is downside is ha i is a mos firs order accurae. If we le (3.3) ω =sup α R we can rewrie (3.3) as v n+ j vj n ( v n j+ v n ) j + H Δ 2Δ H (α) Δ, 2 ( v n j+ 2vj n = ω + ) vn j Δ 2, which is a discreized version of he Hamilon-Jacobi equaion wih an arificial viscosiy erm added, and ω allows he arificial viscosiy o scale wih he discreizaion. The a poseriori error esimae (2.) is based on he noion of viscosiy soluions and so requires he comparison beween coninuous funcions. We choose v o be he piecewise bilinear inerpolan of he numerical soluion given by he above scheme a he poins ( j, n ). We hen sample a he poins ( j+/2, n+/2 ), where j+/2 =( j+ + j )/2 and n+/2 =( n+ + n )/2. A he poins ( j+/2, n+/2 ) we have v( j+/2, n+/2 )= ( v n+ j+ 4 + vn+ j + vj+ n + vj n ), v ( j+/2, n+/2 )= ( v n+ j+ vn+ j + vj+ n vj n ), 2Δ v ( j+/2, n+/2 )= 2Δ ( v n+ j+ vn j+ + v n+ j vj n Choosing he compuaional domain D h as he ses of poins ( j+/2, n+/2 ) ha lie inside D, i.e., defining D h as D h = { ( j+/2, n+/2 )forj, n Z ( j+/2, n+/2 ) D } leads o well-defined derivaives of v a he poins where he soluion is sampled. Hence, a hese poins he semi-differenials are boh nonempy, and p =(p,p )is eacly he differenial of v, i.e., D v( j+/2, n+/2 )=D + v( j+/2, n+/2 )=(v,v )( j+/2, n+/2 ). As he compuaional domain Ω h,t over which we evaluae he L -norm of he difference beween he viscosiy soluion u and he approimaion v when we calculae he effeciviy inde, we ake he union of he ses of poins {( j,t) j Ω} and { ( j+/2,t) j+/2 Ω }. Noe ha calculaing he upper bound for he arificial diffusion coefficien ω in he monoone scheme involves finding he supremum of H (α) over all α R. I is easily seen ha for problems (L)-(L2) and (NC), sup α R H (α) is equal o.75 and, respecively. For problem (C), however, sup α R H (α) is no finie. ).

15 ERROR ESTIMATES FOR HAMILTON-JACOBI EQUATIONS 2 In his case, we ake ω =sup R H (u ()) Δ =Δ. 2 In he numerical eperimens we obain H ( v ( j+/2, n+/2 )) < 2 for all he poins from he compuaional domain, so ha he monooniciy condiion is preserved and he above choice for ω is jusified. Also, recall ha in one space dimension, he parameer V used in he deerminaion of he region enclosed by he rapezoid Q T is given by V = sup p,q R p q H(p ) H(q ), p q where he righ-hand side is assumed o be finie. V is equal o and.75 for problems (L)-(L2) and (NC), respecively. In he numerical eperimens for problem (C) we ake V = sup ( j+/2, n+/2 ) D per h H ( v ( j+/2, n+/2 )), where D per h is he se of poins ( j+/2, n+/2 ) ha lie inside Ω per (,T), i.e., D per h = { ( j+/2, n+/2 )forj, n Z ( j+/2, n+/2 ) Ω per (,T) } Approimaion of he domain Γ Ω,T. To approimae he domain Γ Ω,T inroduced in Remark 5 righ afer our main heoreical resul, Theorem 2.2, we proceed as suggesed herein as follows. Consider he erminal-value problem (3.32) φ + H (u ) φ = for(, ) R (,T), { for Ω, φ(, T )= for / Ω, where Ω R and u is he viscosiy soluion of any of our es problems. We solve his problem backwards using a monoone scheme on he compuaional domain D per h ha we described above. To simplify he noaion, se y j := j+/2 and s n := n+/2. The numerical approimaion o he soluion of (3.32) a he poins (y j,s n ) will be denoed by φ h (y j,s n )orφ n j. The monoone scheme ha we use o solve (3.32) is [ φ n j = φ n+ j +Δ H (v (y j,s n+ )) φn+ j+ φn+ j φ n+ j+ + ω ] 2φn+ j + φ n+ j φ 2Δ Δ 2, where v is he approimae soluion described in he previous secion and where ω φ = sup (y j,s n ) D per h H (v (y j,s n )) Δ. 2 The monoone scheme adds an arificial diffusion erm o equaion (3.32), which causes he numerical approimaion o φ(, ) o be nonzero in a domain ha is larger han he domain Γ Ω,T. To illusrae he effec of he arificial diffusion erm wih a simpler eample, consider he equaion (3.33) ρ = for(, ) R (,T),

16 22 B. COCKBURN, I. MEREV, AND J. QIAN wih he erminal condiion { for, ρ(, T )= for >. Trivially, he soluion is ρ(, ) =ρ(, T ) for all (,T). For κ>, he soluion o (3.34) ρ d = κρ d for (, ) R (,T), wih he same erminal condiion, however, is given by ( ) ρ d (, ) = 2 2 Erf, where Erf() = 2 e p2 dp. 4κ(T ) π Thus, he soluion o he equaion wih he added diffusion erm is monoonically decreasing and nonzero for all <<. Therefore, if we wan o approimae he sub-domain of R, in which he soluion o (3.33) is nonzero by considering he soluion o (3.34), we have o ignore he sub-domain of R, in which he soluion o (3.34) is relaively small. We define our approimaion o he domain Γ Ω,T denoe his approimaion by Γ h,ω,t. A each ime level n of D per h r n = ρ d (ω φ,s n ) / sup ρ d (y, s n )=ρ d (ω φ,s n ) y R and define Γ h,ω,t as { Γ h,ω,t = in a similar fashion. Le us we le (y j,s n )forj, n Z φ h (y j,s n ) >r n sup φ h (y,s n ), (y,s n ) D per h y The fac ha he mehod described above approimaes he domain Γ Ω,T reasonably well in pracice is illusraed by our numerical eperimens Numerical resuls The global a poseriori error esimae. We show he resuls for he compuaional effeciviy indees for smooh soluions of problems (L), (C), and (NC) in Figures 3(a),(c),(e). We see ha in each of he hree problems, he monoone scheme converges linearly, as epeced, and ha he compuaional effeciviy inde is independen of he mesh size Δ. In all of hese cases, he values of he parameers ɛ and ɛ ha minimize he nonlinear funcional of he error esimae are equal o, in agreemen wih our discussion in secion 2. The resuls when he eac soluions of our es problems are nonsmooh are shown in Figures 3(b),(d),(f). We can see ha even for a nonsmooh viscosiy soluion he compuaional effeciviy inde for a linear Hamilonian is nearly opimal. For problems (C) and (NC), he compuaional effeciviy indees increase faser as we decrease he mesh size Δ. Even in hese cases, however, he compuaional effeciviy indees remain relaively small in magniude and are proporional o log(δ). In Table 2 we show he opimal values of he auiliary parameer ɛ =(ɛ,ɛ )for problems (L2), (C), and (NC), when he viscosiy soluions are nonsmooh. In he hird column, we show on how many values of ɛ =(ɛ,ɛ ) he nonlinear funcional of he error esimae has o be evaluaed before hiing is minimum. The las column shows he raio of he opimal values of he auiliary parameers o he larges values ha we allow hem o have in he se (3.24). A reasonable choice for }.

17 ERROR ESTIMATES FOR HAMILTON-JACOBI EQUATIONS log(δ) 3 log(δ) (a) Problem (L): smooh soluion (b) Problem (L2): nonsmooh soluion log(δ) 3 log(δ) (c) Problem (C): smooh soluion (d) Problem (C): nonsmooh soluion log(δ) 3 log(δ) (e) Problem (NC): smooh soluion (f) Problem (NC): nonsmooh soluion Figure 3. Compuaional effeciviy inde as a funcion of log(δ). he auiliary parameers would resul in a relaively sable raio, as is he case for all es problems. As we menioned earlier, he paraboloid es can be used o deec he locaion of he kinks in he eac soluions. This fac is illusraed in Figure 4. We show he pahs of he kinks in he viscosiy soluions along wih poins from he compuaional domains D per h ha fail he paraboloid es when he mesh size Δ =2/28. Noe ha even hough he viscosiy soluion of problem (L2) is nonsmooh for all [,T], we can capure he locaion of he kink for a cerain period only. The reason is ha he arificial diffusion erm in he monoone scheme causes he approimae soluion o smooh ou as ime increases, and afer a cerain poin in ime no poins fail he paraboloid es near he kink in he viscosiy soluion. This can be overcome by considering a finer mesh. A ime T =, he kink in he viscosiy soluion of problem (L2) is a =. For es problem (C), he locaion of he kink in he viscosiy soluion a ime T =.4 is a =.6, and for es problem (NC), he kinks in he viscosiy soluion a

18 24 B. COCKBURN, I. MEREV, AND J. QIAN Table 2. Parameer ɛ =(ɛ,ɛ )fornonsmooh eac soluions. Problem 2/Δ (T/Δ) opimal (ɛ,ɛ ) ɛ-seps raio (L2) 2 () (.e+,.e+). 4 (2) (4.69e 2, 7.3e 3) (4) (3.28e 2, 4.92e 3) (8) (2.e 2, 3.6e 3) (6) (.52e 2, 2.29e 3) (32) (.5e 2,.58e 3) (64) (7.32e 3,.e 3) (C) 2 () (.e+,.e+). 4 (2) (.e, 4.e 2) (4) (7.5e 2, 3.e 2) (8) (4.38e 2,.75e 2) (6) (3.3e 2,.25e 2) (32) (2.34e 2, 9.38e 3) (64) (.64e 2, 6.56e 3) 2.8 (NC) 2 () (.e+,.e+). 4 (2) (.e+,.e+). 8 (4) (.88e 2, 2.85e 3) (8) (.56e 2, 2.37e 3) (6) (9.38e 3,.42e 3) (32) (7.3e 3,.7e 3) (64) (5.8e 3, 7.72e 4) (a) Problem (L2) (b) Problem (C) (c) Problem (NC); kink (d) Problem (NC); kink 2 Figure 4. Grid poins ha fail he paraboloid es around he kinks when 2/Δ = 28. ime T =.5/π 2 are a =.893 and 2 = Table 3 shows he inerval [a, b] a he firs ime level in he compuaional domain for problem (L2) and he las ime level for problems (C) and (NC) over which he paraboloid es fails, i.e., he inerval conaining he se of poins for which condiion (2.2) is no saisfied.

19 ERROR ESTIMATES FOR HAMILTON-JACOBI EQUATIONS 25 We can see ha as we refine he mesh, more poins fail he paraboloid es, and he inerval [a, b] ges smaller. Noe ha for problem (C) we are able o capure he locaion of he kink even for a relaively coarse mesh. The kinks in he eac soluion of problem (NC) are no as poined as he kink in problem (C) (see Figures 2(d),(f)). Because of his fac, we are no able o capure he locaion of he kinks when we use very coarse meshes. Table 3. Paraboloid es: he inerval [a, b] for which he paraboloid es fails a he firs ime level of he compuaional domain for problem (L2) and he las ime level for problems (C) and (NC); henumberofgridpoinsinheinerval[a, b]. number of poins Problem 2/Δ a b inside [a,b] (L2) (C) (NC) kink (NC) kink The local a poseriori error esimae. In his secion we sudy he efficiency of he local a poseriori error esimae wih Q h,t and Γ h,ω,t as he compuaional domains when applied o problems (L2), (C), and (NC). Tes problem (L2). For problem (L2) we consider he inervals [.5,.], [.,.5], and [.3,.45] a ime T =.5. The viscosiy soluion has a kink inside he inerval [.,.5]. I is smooh inside he inerval [.5,.], bu he region enclosed by is corresponding rapezoid Q T conains a par of he pah of he kink. The viscosiy soluion is smooh inside he inerval [.3,.45], as well as inside is corresponding rapezoid Q T. We show he regions enclosed by he rapezoids Q T corresponding o hese inervals ogeher wih he characerisics ha hi he inervals eacly a heir endpoins and he ses of poins from Γ h,ω,t when Δ =2/6 in Figures 5(a),(c),(e). We see ha he mehod described earlier gives a se of poins Γ h,ω,t ha almos perfecly machesheseofpoinsfromd per h ha lie inside Γ Ω,T. Figures 5(b),(d),(f) show he compuaional effeciviy indees as funcions of log (Δ) when Q h,t and Γ h,ω,t are used as he compuaional domains of dependence. As epeced, for boh compuaional domains he smalles effeciviy indees are obained for he inerval [.3,.45]; he viscosiy soluion is smooh boh inside he inerval a ime T and inside he rapezoid Q T.WhenQ T is used, he error esimae

20 26 B. COCKBURN, I. MEREV, AND J. QIAN.5. 5 log(δ) (a) Ω=[.5,.] log(δ) (b) op: Q h,t ; boom: Γ h,ω,t.5..5 log(δ) (c) Ω=[.,.5].5 log(δ) (d) op: Q h,t ; boom: Γ h,ω,t.5..5 log(δ) log(δ) (e) Ω=[.3,.45] (f) op: Q h,t ; boom: Γ h,ω,t Figure 5. Problem (L2); (a),(c),(e): rapezoid Q T (legs are given by he solid black lines); characerisics ha hi he inerval Ω eacly a he endpoins (dashed lines; lef one coincides wih he lef leg of Q T ); pah of he kink (red line); poins from Γ h,ω,t when Δ =2/6); (b), (d), (e): compuaional effeciviy inde as a funcion of log(δ). has he larges compuaional effeciviy inde for he inerval [.5,.], where he viscosiy soluion is smooh, bu whose corresponding Q T conains a par of he pah of he kink. Replacing Q h,t by Γ h,ω,t significanly improves he efficiency of he error esimae in his case. For boh compuaional domains, he compuaional effeciviy inde remains relaively small and close o for he inerval [.,.5], where he viscosiy soluions has a disconinuous derivaive. Table 4 conains informaion abou he opimal values of he auiliary parameers ɛ and ɛ for boh compuaional domains Q h,t and Γ h,ω,t. The raio of he opimal auiliary parameers o he maimum values allowed by he se (3.24) remains sable as he mesh size decreases by a few orders of magniude, confirming ha we have suiably chosen he se for he parameers. As epeced, he opimal

21 ERROR ESTIMATES FOR HAMILTON-JACOBI EQUATIONS 27 values of he auiliary parameers ɛ and ɛ are when he eac soluion is smooh inside he inerval Ω and he domains Q T and Γ Ω,T. Table 4. Parameer ɛ =(ɛ,ɛ ) for problem (L2). Compuaional Inerval domain 2/Δ (T/Δ) opimal (ɛ,ɛ ) ɛ-seps raio [-.5,.] Q h,t 8 (4) (3.28e 2, 4.92e 3) (8) (2.e 2, 3.6e 3) (6) (.52e 2, 2.29e 3) (32) (.5e 2,.58e 3) (64) (7.32e 3,.e 3) [.,.5] Q h,t 8 (4) (3.28e 2, 4.92e 3) (8) (2.e 2, 3.6e 3) (6) (.52e 2, 2.29e 3) (32) (.5e 2,.58e 3) (64) (7.32e 3,.e 3) [.,.5] Γ h,ω,t 8 (4) (3.28e 2, 4.92e 3) (8) (2.e 2, 3.6e 3) (6) (.52e 2, 2.29e 3) (32) (.5e 2,.58e 3) (64) (7.32e 3,.e 3) Tes problem (C). For problem (C) we consider he inervals [.9,.75], [.7,.55], and [.,.5] a T =.4. The viscosiy soluion has a disconinuous derivaive in he inerval [.7,.55]. In [.9,.75], he viscosiy soluion is smooh; however, he region enclosed by is corresponding Q T conains a par of he pah of he kink in he viscosiy soluion. The soluion is smooh inside he inerval [.,.5], as well as inside he rapezoid Q T. We show he regions enclosed by he rapezoids Q T corresponding o hese inervals and he ses of poins from Γ h,ω,t when Δ =2/6 in Figures 6(a),(c),(e). Again noe ha he se of poins Γ h,ω,t maches he se of poins from D per h ha lie inside Γ Ω,T almos perfecly for he inervals [.9,.75] and [.7,.55] and reasonably well for he inerval [.,.5]. We show he resuls for he compuaional effeciviy indees in Figures 6(b),(d),(f). The lowes compuaional effeciviy indees are obained for he inerval [.,.5] for boh domains, as epeced. When Q h,t is used as he compuaional domain, he error esimae has he larges compuaional effeciviy inde for he inerval [.9,.75], where he viscosiy soluion is smooh, bu whose corresponding Q T conains a par of he pah of he kink. The improvemen in he compuaional effeciviy inde for his inerval when we replace Q h,t by Γ h,ω,t as he compuaional domain is significan. The resuls for he inerval [.7,.55] ha conains he kink in he viscosiy soluion are similar for boh compuaional domains. The compuaional effeciviy inde for his inerval is larger compared o he compuaional effeciviy indees for he inervals where he viscosiy soluion is smooh; however, i is sill proporional o log (Δ). Table 5 shows he opimal values of he auiliary parameers ɛ and ɛ. Tes problem (NC). For problem (NC) we consider five differen inervals for which we compue he compuaional effeciviy inde when T =.5/π 2. The viscosiy soluion has a disconinuous derivaive in he inervals [.95,.8] and [.5,.3]. In he inervals [.88,.73] and [.8,.23], he viscosiy soluion is smooh; however, he regions enclosed by heir corresponding rapezoids Q T conain pars of he pahs of he wo kinks. Finally, he viscosiy soluion is smooh inside he inerval [.3,.5], as well as inside is corresponding Q T. We show he

22 28 B. COCKBURN, I. MEREV, AND J. QIAN log(δ) log(δ) (a) Ω=[.9,.75] (b) op: Q h,t ; boom: Γ h,ω,t log(δ) (c) Ω=[.7,.55] log(δ) (d) op: Q h,t ; boom: Γ h,ω,t log(δ) log(δ) (e) Ω=[.,.5] (f) op: Q h,t ; boom: Γ h,ω,t Figure 6. Problem (C); (a),(c),(e): rapezoid Q T (legs are given by he solid black lines); characerisics ha hi he inerval Ω eacly a he endpoins (dashed lines); pah of he kink (red line); poins from Γ h,ω,t when Δ =2/6); (b), (d), (e): compuaional effeciviy inde as a funcion of log(δ). regions enclosed by he rapezoid Q T and he ses of poins from Γ h,ω,t for hese inervals in Figures 7(a),(c),(e),(g),(i). Figures 7(b),(d),(f),(h),(j) show he compuaional effeciviy indees as funcions of log(δ). The resuls are similar o he resuls for es problem (C). As epeced, he smalles effeciviy inde is obained in he cases when he viscosiy soluion is smooh in boh he inerval of consideraion and inside is compuaional domain of dependence. The compuaional effeciviy indees for he inervals [.95,.8] and [.5,.3] ha conain he kinks in he viscosiy soluion are proporional o log(δ) and remain reasonably small as we refine he mesh for boh compuaional domains. The larges effeciviy indees are again obained for he inervals where he viscosiy soluion is smooh, bu whose corresponding

23 ERROR ESTIMATES FOR HAMILTON-JACOBI EQUATIONS 29 rapezoids Q T conain pars of he pah of he kinks in viscosiy soluion, i.e., he inervals [.88,.73] and [.8,.23]. Replacing Q h,t by Γ h,ω,t for hese inervals resuls in significanly lower compuaional indees ha remain close o hroughou he large variaion of he sizes of Δ and Δ. Table 5. Parameer ɛ =(ɛ,ɛ ) for problem (C). Compuaional Inerval domain 2/Δ (T/Δ) opimal (ɛ,ɛ ) ɛ-seps raio [-.9, -.75] Q h,t 8 (4) (6.25e 2, 2.5e 2) (8) (4.38e 2,.75e 2) (6) (3.3e 2,.25e 2) (32) (2.34e 2, 9.38e 3) (64) (.64e 2, 6.56e 3) 2.8 [-.7,.55] Q h,t 8 (4) (7.5e 2, 3.e 2) (8) (4.38e 2,.75e 2) (6) (3.3e 2,.25e 2) (32) (2.34e 2, 9.38e 3) (64) (.64e 2, 6.56e 3) 2.8 [-.7, -.55] Γ h,ω,t 8 (4) (7.5e 2, 3.e 2) (8) (4.38e 2,.75e 2) (6) (3.3e 2,.25e 2) (32) (2.34e 2, 9.38e 3) (64) (.64e 2, 6.56e 3) 2.8 Table 6. Parameer ɛ =(ɛ,ɛ ) for problem (NC). Compuaional Inerval domain 2/Δ (T/Δ) opimal (ɛ,ɛ ) ɛ-seps raio [-.95, -.8] Q h,t 8 (4) (2.5e 2, 3.8e 3) (8) (.56e 2, 2.37e 3) (6) (.9e 2,.66e 3) (32) (7.8e 3,.9e 3) (64) (5.47e 3, 8.3e 4) 4.48 [-.88, -.73] Q h,t 8 (4) (.e+,.e+). 6 (8) (.56e 2, 2.37e 3) (6) (.9e 2,.66e 3) (32) (7.3e 3,.7e 3) (64) (5.8e 3, 7.72e 4) 3.45 [.5,.3] Q h,t 8 (4) (.e+,.e+). 6 (8) (.e+,.e+). 32 (6) (.e+,.e+). 64 (32) (7.3e 3,.7e 3) (64) (5.8e 3, 7.72e 4) 3.45 [-.95, -.8] Γ h,ω,t 8 (4) (2.5e 2, 3.8e 3) (8) (.56e 2, 2.37e 3) (6) (.9e 2,.66e 3) (32) (7.8e 3,.9e 3) (64) (5.47e 3, 8.3e 4) 4.48 [.5,.3] Γ h,ω,t 8 (4) (.e+,.e+). 6 (8) (.e+,.e+). 32 (6) (.e+,.e+). 64 (32) (7.3e 3,.7e 3) (64) (5.8e 3, 7.72e 4) 3.45 Table 6 conains he informaion abou he opimal values of he auiliary parameer ɛ =(ɛ,ɛ ). The opimal values for ɛ and ɛ in he cases when he viscosiy soluion is smooh inside he inervals and inside he compuaional domains of dependence are, as epeced. Noe ha, even hough he region enclosed by he rapezoid Q T corresponding o he inerval [.8,.23] conains a par of he pah of he kink in he viscosiy soluion, he values of he auiliary parameers are equal o. The reason is ha he kink inside Q T is relaively mild (see Figure 2(f)).

24 2 B. COCKBURN, I. MEREV, AND J. QIAN.5. 5 log(δ) log(δ) (a) Ω=[.95,.8] (b) op: Q h,t ; boom: Γ h,ω,t log(δ) log(δ) (c) Ω=[.88,.73] (d) op: Q h,t ; boom: Γ h,ω,t log(δ) log(δ) (e) Ω=[.3,.5] (f) op: Q h,t ; boom: Γ h,ω,t.5 2. log(δ) log(δ) (g) Ω=[.8,.23] (h) op: Q h,t ; boom: Γ h,ω,t log(δ) log(δ) (i) Ω=[.5,.3] (j) op: Q h,t ; boom: Γ h,ω,t Figure 7. Problem (NC); (a),(c),(e): rapezoid Q T (legs are given by he solid black lines); characerisics ha hi he inerval Ω eacly a he endpoins (dashed lines); pah of he kink (red line); poins from Γ h,ω,t when Δ =2/6); (b), (d), (e): compuaional effeciviy inde as a funcion of log(δ).

25 ERROR ESTIMATES FOR HAMILTON-JACOBI EQUATIONS 2 For he cases involving a kink in he eac soluion, he raio of he opimal auiliary parameers o he maimum values ha are allowed in he se (3.24) remains relaively sable as he mesh ges refined, indicaing ha his choice for he se of values for he parameers ɛ and ɛ is reasonable. 4. Concluding remarks In his paper, we obain a generalizaion of he global a poseriori error esimae for ime-dependen Hamilon-Jacobi equaions inroduced in [3] ha gives an upper bound for he difference a ime T beween he viscosiy soluion u of he Cauchy problem (.), (.2) and a coninuous funcion v for any sub-domain of R d. This allows us o define he firs local a poseriori error esimae for ime-dependen Hamilon-Jacobi equaions. We provide numerical eperimens sudying he efficiency of his local error esimae and he global a poseriori error esimae obained in [3]. The numerical eperimens confirm ha even in he difficul cases of nonlinear Hamilonians, when he viscosiy soluion displays kinks, he error esimaes applied o a monoone scheme produce effeciviy indees ha remain relaively small as he discreizaion parameers vary in several orders of magniude. This shows ha hese a poseriori error esimaes could be effecively used in adapive algorihms wih rigorous error conrol for ime-dependen Hamilon-Jacobi equaions, similar o he algorihms proposed in [5], [6], and [7] for seady-sae Hamilon-Jacobi equaions. Two opics consiue he subjec of ongoing research. The firs is a heoreical jusificaion of he use of he se Γ Ω,T insead of he rapezoid Q T in he a poseriori error esimae. The second is he eension of he esimaes o high-order schemes. Noe ha he use of he a poseriori error esimae wih formally high-order numerical schemes is likely o give rise o a subopimal effeciviy inde jus as for he seady-sae case; see [2]. In [5], he esimae obained in [2] had o be suiably modified before applied o high-order schemes like disconinuous Galerkin mehods. I is hus reasonable o epec ha a similar modificaion should be carried ou in he ime-dependen case also. References [] R. Abgrall, Numerical discreizaion of he firs-order Hamilon-Jacobi equaion on riangular meshes, Comm. Pure Appl. Mah. 49 (996), MR44589 (98d:652) [2] S. Alber, B. Cockburn, D. French, and T. Peerson, A poseriori error esimaes for general numerical mehods for Hamilon-Jacobi equaions. Par I: The seady sae case, Mah. Comp. 7 (22), MR (22m:657) [3], A poseriori error esimaes for general numerical mehods for Hamilon-Jacobi equaions. Par II: The ime-dependen case, Finie Volumes for Comple Applicaions, (R. Herbin and D. Kröner, eds.), vol. III, Hermes Penon Science, June 22, MR274 [4] T. Barh and J. Sehian, Numerical schemes for Hamilon-Jacobi and level se equaions on riangular domains, J.Compu.Phys.45 (998), 4. MR6442 (99d:65277) [5] Y. Chen and B. Cockburn, An adapive high order disconinuous Galerkin mehod wih error conrol for ime Hamilon-Jacobi equaions. Par I: The one-dimensional seady sae case, J. Compu. Phys. 226 (27), MR (28k:65239) [6] B. Cockburn and B. Yenikaya, An adapive mehod wih rigorous error conrol for he Hamilon-Jacobi equaions. Par I: The one-dimensional seady sae case, Appl. Numer. Mah. 52 (25), MR269 (25i:652)

26 22 B. COCKBURN, I. MEREV, AND J. QIAN [7], An adapive mehod wih rigorous error conrol for he Hamilon-Jacobi equaions. Par II: The wo-dimensional seady sae case, J. Compu. Phys. 29 (25), MR2599 (26b:6585) [8] M. G. Crandall, L. C. Evans, and P. L. Lions, Some properies of viscosiy soluions of Hamilon-Jacobi equaions, Trans. Amer. Mah. Soc. 282 (984), MR7322 (86a:353) [9] M.G.CrandallandP.L.Lions,Two approimaions of soluions of Hamilon-Jacobi equaions, Mah.Comp.43 (984), 9. MR74492 (86j:652) [] M. Falcone and R. Ferrei, Discree ime high-order schemes for viscosiy soluions of Hamilon-Jacobi-Bellman equaions, Numer. Mah. 67 (994), MR2695 (95d:4945) [] C. Hu and C.-W. Shu, A disconinuous Galerkin finie elemen mehod for Hamilon-Jacobi equaions, SIAM J. Sci. Compu. 2 (999), MR78679 (2g:6595) [2] H. Ishii, Uniqueness of unbounded viscosiy soluion of Hamilon-Jacobi equaions, Indiana Univ. Mah. Journal 33 (984), MR75656 (85h:3557) [3] S. Osher and C.-W. Shu, High-order essenially nonoscillaory schemes for Hamilon-Jacobi equaions, SIAMJ.Numer.Anal.28 (99), MR446 (92e:658) [4] J. Qian, Approimaions for viscosiy soluions of Hamilon-Jacobi equaions wih locally varying ime and space grids, SIAM J. Numer. Anal. 26, MR22644 (27b:4955) [5] P. E. Souganidis, Approimaion schemes for viscosiy soluions of Hamilon-Jacobi equaions, J.Diff.Eqns.59 (985), 43. MR8385 (86k:3528) [6] G. B. Whiham, Linear and nonlinear waves, Wiley, New York, NY, 974. MR (58:395) School of Mahemaics, Universiy of Minnesoa, 26 Church Sree S.E., Minneapolis, Minnesoa address: cockburn@mah.umn.edu School of Mahemaics, Universiy of Minnesoa, 26 Church Sree S.E., Minneapolis, Minnesoa address: merev@mah.umn.edu Deparmen of Mahemaics, Michigan Sae Universiy, Eas Lansing, Michigan address: qian@mah.msu.edu