Symmtric Intrior Pnalty Galrkin Mthod for Elliptic Problms Ykatrina Epshtyn and Béatric Rivièr Dpartmnt of Mathmatics, Univrsity of Pittsburgh, 3 Thackray, Pittsburgh, PA 56, U.S.A. Abstract This papr prsnts computabl lowr bounds of th pnalty paramtrs for stabl and convrgnt symmtric intrior pnalty Galrkin mthods. In particular, w driv th xplicit dpndnc of th corcivity constants with rspct to th polynomial dgr and th angls of th msh lmnts. Numrical xampls in all dimnsions and for diffrnt polynomial dgrs ar prsntd. W invstigat th numrical ffcts of loss of corcivity. Ky words: Corcivity, stabl v. unstabl intrior pnalty, lliptic problms Introduction Th Symmtric Intrior Pnalty Galrkin (SIPG) mthod for lliptic problms was first introducd in th lat svntis by Douglas and Dupont [9], Whlr [] and Arnold [,] and was rvivd mor rcntly as a popular discontinuous Galrkin mthod. Som of th gnral attractiv faturs of th mthod ar th local and high ordr of approximation, th flxibility du to local msh rfinmnt and th ability to handl unstructurd mshs and discontinuous cofficints. Mor spcific proprtis includ th optimal rror stimats in both th H and L norms and th rsulting symmtric linar systms asily solvd by standard solvrs for symmtric matrics (such as conjugat gradint). Th analysis and application of SIPG to a wid rang of problms can b found in th litratur: a non-xhaustiv list is givn in [4,5,7,,6,8,9,4] and th rfrncs hrin. This rsarch is partially fundd by NSF-DMS 5639.
Th SIPG mthod is obtaind by intgrating by parts on ach msh lmnt, and summing ovr all lmnts. Two stabilization trms ar thn addd: a symmtrizing trm corrsponding to fluxs obtaind aftr intgration by part, and a pnalty trm imposing a wak continuity of th numrical solution. It is wll known that thr xists a thrshold pnalty abov which th bilinar form is corciv and th schm is stabl and convrgnt. Anothr rlatd discontinuous Galrkin mthod is th non-symmtric intrior pnalty Galrkin (NIPG) mthod [7,]: this mthod diffrs from th SIPG mthod by only on sign: th symmtrizing trm is addd instad of bing subtractd. On on hand, th loss of symmtry in th schm givs an immdiat corcivity of th bilinar form; th NIPG schm is stabl and convrgnt for any valu of th pnalty. On th othr hand, optimal rror stimats in th L norm cannot b provd via th standard Nitsch lift. As of today, this rmains an opn problm. Th objctiv of this work is to driv rigorous computabl bounds of th thrshold pnalty that would yild a stabl and convrgnt SIPG. W considr a gnral scond ordr lliptic problm on a domain in any dimnsion, subdividd into simplics. Our main rsult is an improvd corcivity rsult. In particular, w show that th constant of corcivity dpnds on th polynomial dgr and th smallst sin θ ovr all angls θ in th triangular msh in D or ovr all dihdral angls θ in th ttrahdral msh in 3D. W also invstigat th ffcts of th pnalty numrically and xhibit unstabl oscillatory solutions for pnalty valus blow th thrshold pnalty. Our rsults also apply to th incomplt intrior pnalty Galrkin mthod [8], that diffrs from SIPG and NIPG in th fact that th symmtrizing stabilizing trm is rmovd. For this mthod, th rror analysis in th nrgy norm is idntical to th analysis of th SIPG mthod. Th outlin of th papr is as follows: th modl problm and schm ar prsntd in Sction. Sction 3 contains th improvd corcivity thorms. Sction 4 shows numrical xampls in all dimnsions that support our thortical rsults. Som conclusions follow. Modl Problm and Schm Lt Ω b a domain in R d, d =,, 3. Lt th boundary of th domain Ω b th union of two disjoint sts Γ D and Γ N. W dnot n th unit normal vctor to ach dg of Ω xtrior of Ω. For f givn in L (Ω), u D givn in H (Γ D ) and u N givn in L (Γ N ), w considr th following lliptic problm:
(K u) + αu = f in Ω, () u = u D on Γ D, () K u n = u N on Γ N. (3) Hr, th function α is a nonngativ scalar function and K is a matrix-valud function K = (k ij ) i,j d that is symmtric positiv dfinit, i.. thr xist two positiv constants k and k such that x R d, k x T x x T Kx k x T x. (4) W can assum that th problm ()-(3) has a uniqu solution in H (Ω) whn Γ D > or whn α. On th othr hand, whn Ω = Γ N and α =, problm ()-(3) has a solution in H (Ω) which is uniqu up to an additiv constant, providd Ω f = Ω g. Lt T h = {E} E b a subdivision of Ω, whr E is an intrval if d =, a triangl if d =, or a ttrahdron if d = 3. Lt whr h E is th diamtr of E. h = max E T h h E, Lt p b a positiv intgr. Dnot by P p (E) th spac of polynomials of total dgr lss than p on th lmnt E. Th finit lmnt subspac is takn to b D p (T h ) = {v h L (Ω) : E T h v h E P p (E)}. W not that thr ar no continuity constraints on th discontinuous finit lmnt spacs. In what follows, w will dnot by O th L norm ovr th domain O. W now prsnt th schm. For radibility purposs, w sparat th ondimnsional cas from th highr dimnsional cas.. SIPG in On Dimnsion Assuming that Ω = (a, b), w can writ th subdivision: T h = {I n = (x n, x n+ ) : n =,..., N } with x = a and x N = b. W assum that Γ D = {a, b} and thus Γ N =. If w dnot v(x + n ) = lim ε + v(x n + ε) and v(x n ) = lim ε + v(x n ε), w can dfin 3
th jump and avrag of v at th ndpoints of I n : n =,..., N, [v(x n )] = v(x n ) v(x + n ), {v(x n )} = (v(x n ) + v(x + n )), [v(x )] = v(x + ), {v(x )} = v(x + ), [v(x N)] = v(x N ), {v(x N)} = v(x N ). Th SIPG finit lmnt mthod for problm ()-(3) is thn : find u h in D p (T h ) such that : v h D p (T h ), A(u h, v h ) = L(v h ), (5) whr th bilinar form A and linar form L ar dfind by: A(w, v) = + N n= N ( σ n n= N n= L(v) = xn+ (K(x)w (x)v (x) + αw(x)v(x))dx + σ x n I [w(x )][v(x )] I n+ + ) [w(x n )][v(x n )] + σ N I n I N [w(x N)][v(x N )] N {K(x n )w (x n )}[v(x n )] b a n= {K(x n )v (x n )}[w(x n )], (6) f(x)v(x)dx + K(a)v (a)u D (a) K(b)v (b)u D (b) + σ I v(a)u D(a) + σ N I N v(b)u D(b), (7) whr {σ n } n ar ral positiv pnalty paramtrs dfind on ach subintrval I n indpndntly. W dnot by σ > th minimum of all σ n. Th nrgy norm associatd to A is: v h D p (T h ), v h E = + σ N I [v(x )] + ( σ n n= ( N n= xn+ x n I n+ + I n (K(x)(v h (x)) + α(x)(v h (x)) )dx ) [v(x n )] + σ ) N /. I N [v(x N)] (8). SIPG in High Dimnsions Lt Γ h b th st of intrior dgs in D (or facs in 3D) of th subdivision T h. With ach dg (or fac), w associat a unit normal vctor n. If is on th boundary Ω, thn n is takn to b th unit outward vctor to Ω. W now dfin th avrag and th jump for w: = E E, {w} = (w E ) + (w E ), [w] = (w E ) (w E ), = E Ω, {w} = (w E ), [w] = (w E ). 4
Th gnral SIPG variational formulation of problm ()-(3) is: find u h in D p (T h ) such that: v h D p (T h ), A(u h, v h ) = L(v h ), (9) whr th bilinar form A and linar form L ar dfind by: L(v) = A(w, v) = Ω E T h Γ h Γ D fv Γ D E K w v + {K w n }[v] Ω αwv + Γ h Γ D (K v n )u D + Γ D Γ h Γ D σ β [w][v] {K v n }[w], () σ β vu D + Γ N vu N. () Th pnalty paramtr σ is a positiv constant on ach dg (or fac) and w dnot by σ > th minimum of all σ. Th paramtr β > is a global constant that, in gnral, is chosn to b on. If β >, thn th SIPG mthod is said to b suprpnalizd. Th nrgy norm associatd to A is: ( v h D p (T h ), v h E = E T E h K( v h ) + Ω αv h + Γ h Γ D σ β ) [v h ]. ().3 Error Analysis W rcall th wll-known rsults about th schms (5) and (9). Lmma Consistncy. Th xact solution of ()-(3) satisfis th discrt variational problm (5) in on dimnsion and (9) in two or thr dimnsions. Lmma Corcivity. Thr xists a pnalty σ such that for any σ > σ w hav v h D p (T h ), A(v h, v h ) C v h E, for som positiv constant C indpndnt of h. Lmma 3 Continuity. Thr xists a constant C such that v h, w h D p (T h ), A(v h, w h ) C v h E w h E. Thorm 4 Error stimats. Lt u H p+ (Ω) b th xact solution of ()- (3). Assum that th corcivity lmma holds tru. In addition, assum that 5
β. Thn, thr is a constant C indpndnt of h, but dpndnt of C, such that u u h E Ch p u H p+ (Ω). Ths rsults ar provd by using standard trac inqualitis [6] and thy can b found for xampl in [,3,3]. Th aim of this work is to dtrmin xactly what is th valu σ that would guarant th corcivity. W also obtain a prcis xprssion for both corcivity and continuity constants C, C. W thn show numrically that for pnalty valus lowr than σ, unstabl solutions could occur. 3 Improvd Corcivity and Continuity Lmmas W will considr ach dimnsion sparatly as th dtails of th proofs diffr. 3. Estimation of σ in On Dimnsion Thorm 5 Lt ε = k. For any ε >, dfin σ n (ε) = k (p+) ε n =,..., N, 3k (p+) 4ε n =, N. (3) Thn for any < ε < ε, if σ n > σn (ε) for all n, thr is a constant < C (ε) <, indpndnt of h, such that v h D p (T h ), A(v h, v h ) C (ε) v h E. Morovr, an xprssion for C (ε) is: C (ε) = min{ ε, 3k (p + ), k (p + ),..., k (p + ), 3k (p + ) }. k 4εσ εσ εσ N 4εσ N Proof Choosing w = v in (6) yilds xn+ N N A(v, v) = K(x)(v (x)) dx {K(x n )v (x n )}[v(x n )] + σ n= x n n= I [v(x )] N ( + σ n n= I n+ + ) [v(x n )] + σ N I n I N [v(x N)]. (4) 6
It suffics to bound th trm N n= {K(x n )v (x n )}[v(x n )] and obtain som rstrictions on th pnalty paramtrs {σ n } n for th corcivity to hold. Lt us first considr on intrior point x n. By dfinition of th avrag and th proprty (4), w hav {K(x n )v (x n )} K(x n )v (x n ) + K(x+ n )v (x + n ) k ( v (x n ) + v (x + n ) ). (5) For any intrval I = (s, t), th following improvd invrs trac inquality holds []: v h P p (I), v h (s) p + I v h I. (6) Hnc using (6) w can bound v (x n ) and v (x + n ) : v (x n ) p + xn x n v (x) (xn,x n), v (x + n ) p + xn+ x n v (x) (xn,x n+ ). Using ths bounds w obtaind for th intrior point x n of th subdivision: {K(x n )v (x n )}[v(x n )] k (p + ) Lt us considr now th boundary nods x and x N : ( v ) (x) (xn,x n) + v (x) (xn,x n+ ) [v(x n )]. xn x n xn+ x n {K(x )v (x )}[v(x )] K(x )v (x )[v(x )] (7) k (p + ) v (x) (x,x ) (x x ) [v(x )], (8) {K(x N )v (x N )}[v(x N )] K(x N )v (x N )[v(x N )] k (p + ) v (x) (xn,x N ) (x N x N ) [v(x N)]. (9) Combining th bounds abov givs: N {K(x n )v (x n )}[v(x n )] k ( (p + ) v [v(x )] (x) (x,x n= ) x x + N v ( [v(x n )] + [v(x n )] ) (x) (xn,x n) + v [v(x N )] (x) (xn,x xn x N ) ). n xn x N n= 7
Aftr application of discrt Cauchy-Shwarz s inquality w hav: N {K(x n )v (x n )}[v(x n )] k N (p + ) ( v (x) (x,x) n= + v (x) (x n,x n) n= + v (x) (x N,x N ) ) ( [v(x )] + x x N n= Application of Young s inquality yilds: ( [v(x n )] + [v(x n )] ) + [v(x N)] ). x n x n x N x N N {K(x n )v (x n )}[v(x n )] ε K / v (x) N ε,i n= k + K / v (x),i n= k n + ε ( K / v (x),i k N + k (p + ) 3 ε 8 I [v(x )] N ( + n= I n + ) [(v(x n )] + 3 ) I n+ 8 I N [v(x N)]. () Hnc using th stimat () w obtain a lowr bound for th bilinar form (4): ( A(v, v) ε ) K(x)v (x) N dx + ( ε ) K(x)v (x) dx k I n= k I n ( + ε ) ) K(x)v (x) dx + (σ 3k (p + ) [v(x )] () k I N 4ε I N + (σ n k (p + ) )( n= ε I n+ + ) [v(x n )] + (σ N 3k (p + ) ) [v(xn )]. I n 4ε I N Lt us dnot ε = k. From () th bilinar form (4) is corciv if : ε < ε () and k (p+) n =,..., N, ε σ n > 3k (p+) n =, N. 4ε This concluds th proof. (3) Similarly, on can show th following improvd continuity constant. Lmma 6 Undr th notation of Thorm 5, th continuity constant C of Lmma 3 is givn by: C = max{ + 3, + 8,..., + 8, + 3, + k (p + ) }. σ σ σ N σ N k 8
3. Estimation of σ in Two Dimnsions In this sction, w dnot θ T th angl such that sin θ T has th smallst valu ovr all triangls. In two dimnsions, it is to b notd that this angl θ T corrsponds to th smallst angl ovr all triangls in th subdivision. W show that th corcivity constant dpnds on θ T. In Sction 4, w outlin a simpl algorithm for computing such angl. Thorm 7 For any ε >, dfin σ (ε) = 5 k ε (p + )(p + ) cot θ T β. (4) Thn for any < ε < k, if σ > σ (ε) for all, thr is a constant < C (ε) <, indpndnt of h, such that An xprssion for C is: C (ε) = v h D p (T h ), A(v h, v h ) C (ε) v h E. min { ε, 5 k Γ h Γ D k εσ (p + )(p + ) cot θ T β }. Proof: Similarly, as in th on-dimnsional cas, w choos w = v in (): A(v, v) = K( v) + αv E T E Ω h {K v n }[v] + σ [v]. (5) Γ h Γ D Γ h Γ D β In ordr to hav corcivity of th bilinar form w nd to bound th trm Γ h Γ D {K v n }[v]. Lt us first considr on intrior dg shard by two triangls E and E. Applying Cauchy-Schwarz inquality w hav: {K v n }[v] {K v n } [v]. (6) Using th dfinition of th avrag and th proprty (4), w hav {K v n } K v n E + K v n E. 9
θ θ 3 E θ 3 3 Fig.. Angls and dgs in a gnric triangl. ) ( K v E + K v E k ) ( v E + v E, (7) so w obtain for th intrior dg : {K v n }[v] k ) ( v E + v E [v]. (8) Similarly, for a boundary dg blonging to th boundary of lmnt E: {K v n }[v] k v E [v]. (9) W now rcall th invrs inquality valid on an dg of a triangl E []: v h P p (E), v h (p + )(p + ) E u E. (3) Hnc in (3) w nd to stimat th ratio, whr is on dg of a E triangl E. For this, w considr a triangl with dgs, and 3. W dnot by θ ij th intrior angl btwn dg i and dg j (s Fig. ).Without loss of gnrality, w assum that = 3. Th ara of th triangl E is givn by th formula: E = i j sin θ ij = 4 3 sin θ 3 + 4 3 sin θ 3 Th lngth of th dg in th triangl E can also b writtn as : Hnc, using th smallst angl θ T = 3 = cos θ 3 + cos θ 3. w hav: E = 4 ( ) cos θ 3 + cos θ 3 sin θ 3 + sin θ 3 So w obtain th following stimat : 4 ( ) cos θ T + cos θ T. sin θ T + sin θ T E 4 cot θ T. (3)
Thn using invrs inquality (3), and th stimat (3) in (8) and (9) w obtain for th intrior dg of th triangl E and E : {K v n }[v] k (p + )(p + ) and for th boundary dg: {K v n }[v] k (p + )(p + ) cot θ T ( v,e + v,e ) [v], (3) cot θ T v E [v]. (33) Combining th bounds abov and using discrt Cauchy-Schwarz s inquality, w obtain: {K v n }[v] k (p + )(p + ) cot θ T Γ h Γ D ( ( v E + v E ) Γ h [v] + ) v E Γ D [v] k (p + )(p + ) cot θ T ( ( v E + v E ) + ) / ( ) v /. E Γ h Γ D Γ h Γ D [v] (34) W now rwrit th first sum ovr dgs as a sum ovr triangls by dcomposing th subdivision into disjoint sts T, T D, T D, T N, T N and T DN. Th st T rprsnts th st of triangls with thr intrior dgs. Th st T D rprsnts th st of triangls with two intrior dgs and on boundary dg of Dirichlt typ. Th st T D rprsnts th st of triangls with on intrior dg and two dgs on th Dirichlt boundary. Th st T N rprsnts th st of triangls with two intrior dgs and on boundary dg of typ Numann. Th st T N rprsnts th st of triangls with on intrior dg and two boundary dgs of typ Numann. Finally, th st T DN rprsnts th st of triangls with on intrior dg, on Numann dg and on Dirichlt dg. Th sum ovr dgs can thn b rwrittn as: ( v E + v E ) + v E = 3 v E + 4 v E Γ h Γ D E T E T D + 5 v E + v E + v E + 3 v E E T D E T N E T N E T DN 5 v E. E T h
Thrfor, by using Young s inquality and th proprty (4), w hav for any positiv ε: {K v n }[v] ε K( v) Γ h Γ D k E T E h + 5 k (p + )(p + ) cot θ T β [v]. (35) ε Γ h Γ D β Thrfor using th stimat (35) w hav th following lowr bound for th bilinar form (5): A(v, v) ( ε ) K v E k E T h + 5 σ ε k (p + )(p + ) cot θ T β [v] Γ h Γ D β. (36) From (36) th bilinar form (5) is corciv if th following two conditions hold: ε < k, (37) σ > 5 k (p + )(p + ) cot θ T β. (38) k This concluds th proof. Corollary 8 Assum that no supr-pnalization is usd, namly β =, thn th stimat is indpndnt of h: < ε < k, σ (ε) = 5 k ε (p + )(p + ) cot θ T. Lmma 9 Undr th notation of Thorm 7, th continuity constant C of Lmma 3 is givn by: C = max { + β, + 5k (p + )(p + ) cot θ T }. Γ h Γ D k σ 3.3 Estimation of σ in Thr Dimnsions Thorm Lt θ T dnot th dihdral angl such that sin θ T has th smallst valu ovr all ttrahdrons in th subdivision. For any ε >, dfin σ(ε) = k 4 ε (p + )(p + 3)h cot θ T β. (39)
d Fig.. A ttrahdral lmnt with facs i. Thn for any < ε < k, if σ > σ (ε) for all facs, thr is a constant < C (ε) <, indpndnt of h, such that An xprssion for C is C (ε) = v h D p (T h ), A(v h, v h ) C v h E. min { ε, k Γ h Γ D k 4εσ (p + )(p + 3) cot θ T β }. Proof: Th proof is similar to th on for th two-dimnsional cas, and thus w will skip som tchnical dtails. W first rcall th invrs inquality in 3D for a ttrahdral lmnt E with fac []: v h P p (E), v h (p + )(p + 3) 3 E v h E. (4) Hr, is th ara of th fac and E is th volum of th ttrahdral lmnt. So as in th cas of th triangl w nd to stimat th ratio. For this, E w fix an lmnt E in T h and w dnot by i, i =,..., 4 th facs of E and by d ij th common dg to facs i and j. W will assum that th fac is dnotd by 4. W also dnot by θ ij th dihdral angl btwn facs i and j. A schmatic is givn in Fig.. Th volum of th ttrahdron is givn by th formula [5]: E = 3 d ij i j sin θ ij, (4) thrfor w can rwrit th volum as: E = ( 3 3 d 4 4 sin θ 4 + 3 d 4 4 sin θ 4 + ) 3 d 34 4 3 sin θ 34 = ( 9 4 sin θ 4 + sin θ 4 + 3 d 4 d 4 Hnc, using th fact that d ij h, w hav : d 34 sin θ 34 E = 4 E = 4 ( ) 9 4 sin θ d 4 4 + sin θ d 4 4 + 3 sin θ d 34 34 ). (4) 3
9 4 4 ( sin θ h 4 + sin θ h 4 + 3 sin θ h 34 9 h 4 ). (43) 4 ( sin θ 4 + sin θ 4 + 3 sin θ 34 ) Th rlation btwn aras of th facs and dihdral angls in gnral ttrahdron is givn by th formula [5]: k = 4 i cos θ ki. (44) i k i= Hnc w hav using (44) in (43) and using dihdral angl θ T E 9 ( ) h cos θ 4 + cos θ 4 + 3 cos θ 34 4 sin θ 4 + sin θ 4 + 3 sin θ 34 dfind abov: 9 ( ) h cos θ T + cos θ T + 3 cos θ T. 4 sin θ T + sin θ T + 3 sin θ T Thrfor w obtain th following stimat for a givn fac in ttrahdral lmnt E: E 9 h cot θ T, (45) which is similar to stimat (3). Using a similar argumnt as in th triangular cas, w obtain for th intrior fac of th ttrahdral lmnt E and E : ) {K v n }[v] k 3(p + )(p + 3) h cot θ T ( v E + v E [v]. 8 (46) and for th boundary fac w hav : {K v n }[v] k 3(p + )(p + 3) h cot θ T v E [v] (47) Thrfor w can stimat now th trm Γ h Γ D {K v n }[v]. W first apply a discrt Cauchy-Schwarz s inquality, thn w dcompos th subdivision into disjoint sts T, T D, T D, T 3D,... as in th D cas. Th gratst cofficint corrsponds to th cas of a ttrahdron with thr Dirichlt boundary facs. Thrfor, it is asy to s that w obtain for any ε > : {K v n }[v] ε Γ h Γ D k E T h E K( v) + 4
k (p + )(p + 3)h cot θ T β [v]. (48) 8ε Γ h Γ D β Thrfor using th stimat (48) w hav th following bound for th bilinar form (5): A(v, v) ( ε ) ( v) E + k E T h K σ 4ε k (p + )(p + 3)h cot θ T β [v] Γ h Γ D β. (49) Corcivity is thn obtaind for ε and σ satisfying th bounds: ε < k, (5) σ > k 4 ε (p + )(p + 3)h cot θ T β. (5) This concluds th proof. Corollary Assum that no supr-pnalization is usd, namly β =, thn th stimat bcoms: σ = k (p + )(p + 3)h cot θ T. 4 k Lmma Undr th notation of Thorm, th continuity constant C of Lmma 3 is givn by: C = max { + β, + Γ h Γ D 4 σ k k (p + )(p + )h cot θ T }. 4 Numrical xampls W now prsnt simpl computations obtaind for th domains Ω, Ω, Ω 3 in D, D and 3D rspctivly. Th xact solutions ar priodic functions dfind by: u (x) = cos(8πx) on Ω = (, ), u (x) = cos(8πx) + cos(8πy) on Ω = (, ), u 3 (x) = cos(8πx) + cos(8πy) + cos(8πz) on Ω 3 = (, ) 3. Th tnsor K is th idntity tnsor. W fix β =. W vary th numbr of lmnts N h in th msh, th polynomial dgr and th pnalty valu (dnotd by σ) that is chosn, for simplicity, constant ovr th whol domain. In ach cas, w prcis th limiting pnalty valu σ = σ (ε ). 5
...3.4.5.6.7.8.9 4 3.5 3.5.5 3 3...3.4.5.6.7.8.9.5...3.4.5.6.7.8.9 Fig. 3. p =, σ =.5: N h = (lft), N h = (cntr), N h = 4 (right)..5.5.5.5.5.5.5.5.5...3.4.5.6.7.8.9...3.4.5.6.7.8.9...3.4.5.6.7.8.9 Fig. 4. p =, σ = 6.5: N h = (lft), N h = (cntr), N h = 4 (right). 4. On-dimnsional Problm W first considr th cas of picwis linars on svral mshs containing, and 4 intrvals rspctivly. In all figurs, th xact solution is drawn as a dashd lin whras th numrical solution is drawn as a solid lin. For a pnalty valu σ =.5 that is smallr than σ = 6, oscillations occur for all thr mshs (s Fig. 3) and th numrical rror is larg. Whn σ > σ, th numrical solution is accurat (s Fig. 4). Th two curvs coincid with ach othr. Th rrors dcras as th msh is rfind according to th thortical convrgnc rat givn in Thorm 4. W rpat th numrical xprimnts with picwis quadratics and picwis cubics. Unstabl solutions ar obtaind for pnalty valus blow th thrshold valu (s Fig. 5 and Fig. 7). Th stabl and convrgnt solutions ar shown in Fig. 6 and Fig. 8. It is intrsting to point that for th unstabl pnalty σ = 3.583, th solution is accurat for th msh with lmnts; howvr larg oscillations occur on mshs with and 4 lmnts. Finally, Fig. 9 corrsponds to a zro pnalty on a coars msh and a vry fin msh: as xpctd, rfining th msh is not nough to rcovr from th loss of corcivity. A mor prcis stimat of th accuracy is givn in Tabl 4.. Th absolut L rror u u h Ω and H rror ( ( (u u h ) E + u u h E)) / ar E T h computd for ach simulation. W also indicat th limiting pnalty valus σn for all n =,..., N. For stabl solutions, w choos pnalty valus that ar gratr than th limiting valu. It is to b notd that whn σ is vry clos to th limiting valu σn, th corcivity constant C is vry clos to zro. In that cas, numrical oscillations could still occur. This poor corcivity proprty is discussd in dtail in []. 6
Tabl Numrical rrors for on-dimnsional simulations. N h p σ n σ n <n<n σ n n=,n L rror H rror 6 5.7794 67.355 6 6.5748.6545.5 6.4784 9.68 6.5 6.367.677.5 6.43 4.65 6.5 6.93 6.879 4.5 6.334 9.763 4 6.5 6.47 3.48.375 36 7.366 3.8899 37 36 7.558 3.8357.375 36 7.6.3 37 36 7.73.5 4.375 36 7.65.474 4 37 36 7 9.35 4.66 3 3.583 64 48. 9.4335 3 65 64 48.8.864 3 3.583 64 48.7.45 3 65 64 48 5.573 4.93 4 3 3.583 64 48.3497 467.898 4 3 65 64 48 3.5977 5.38.5.5.5.5.5.5.5.5.5.5.5...3.4.5.6.7.8.9...3.4.5.6.7.8.9.5...3.4.5.6.7.8.9 Fig. 5. p =, σ =.375: N h = (lft), N h = (cntr), N h = 4 (right). 4. Two-dimnsional Problm W first xplain how to obtain th angl θ T. This angl will giv th largst cos θ, or th largst cot θ ovr all triangl angls θ. For a givn lmnt E, w 7
...3.4.5.6.7.8.9.5.5.5.5.5.5.5.5.5...3.4.5.6.7.8.9...3.4.5.6.7.8.9 Fig. 6. p =, σ = 37: N h = (lft), N h = (cntr), N h = 4 (right)..5.5 6 4.5.5.5.5 4.5...3.4.5.6.7.8.9.5...3.4.5.6.7.8.9 6...3.4.5.6.7.8.9 Fig. 7. p = 3, σ = 3.583: N h = (lft), N h = (cntr), N h = 4 (right)..5.5.8.6.5.5.4..5..4.5.6.8.5...3.4.5.6.7.8.9...3.4.5.6.7.8.9...3.4.5.6.7.8.9 Fig. 8. p = 3, σ = 65: N h = (lft), N h = (cntr), N h = 4 (right). 3 4 3 3 3...3.4.5.6.7.8.9 4...3.4.5.6.7.8.9 Fig. 9. p =, σ = : coars msh N h = (lft) and rfind msh N h = 6 (right). comput a valu cot θ E dfind by: () Comput th lngths of th dgs of E from th vrtics coordinats (x E i, y E i ): ( = x E x E ) + (y E y E ) ) / ( = x E 3 x E ) + (y3 E y E ) ) / ( 3 = x E xe 3 ) + (y E ye 3 ) ) / () Dtrmin th smallst lngth, say i. Dnot th othr two lngths by i and i3. (3) Comput cot θ E : cos θ E = i + i3 i, sin θ E = ( (cos θ E ) ) /, cot θ E = cos θ E i i sin θ E 8
Y X Fig.. Structurd msh with 8 lmnts. Fig.. Exact solution: two-dimnsional viw (lft) and thr-dimnsional viw (right) Th valu cot θ T is th maximum of cot θ E ovr all msh lmnts E. W solv th problm on structurd mshs as shown in Fig.. For this msh, th smallst angl is θ T = π. Th xact solution for rfrnc is shown 4 in Fig.. In Fig. and 3, w first considr polynomial dgr qual to on on a vry fin msh (48 lmnts): th pnalty paramtr taks th valus, 3 and 5. In this cas, th limiting valu is σ = 3. For a pnalty valu abov th limiting valu, no oscillations occur whras for a pnalty valu blow σ, th solution is unstabl. Fig. 4 and 5 show th solution for polynomial dgr on a msh containing 8 lmnts. W thn rfin th msh (5 lmnts) and obtain Fig. 6 and 7. Finally, for th cas of picwis cubic polynomials, th solutions ar shown in Fig. 8 and 9 for a msh containing 3 lmnts, and in Fig. and for a msh containing 8 lmnts. W giv th rror in th L norm for all cass and w also giv th limiting valu σ in Tabl 4.. For a givn pnalty, th rror dcrass as th msh is rfind. Similar conclusions as in th on-dimnsional cas can b mad. For stabl mthod, th rror dcrass with th right convrgnc rat. For unstabl mthod, oscillations may occur. 9
Tabl Numrical rrors for two-dimnsional simulations. N h p σ σ L rror H rror 48 3.6868 7.995783 48 3 3 9.749787.8656 48 5 3 5.3487 5.54996 8 6.784956.398348 8 4.5 6 4.975469.84357 8 6.75758.8489 5 6 5.34755 4.384793 5 4.5 6.744388.636 5 6.7976364.673 3 3 4.647666 6.699869 3 3 5.643677 3.83936 3 3 3 5 4.7347586.66337 8 3 7.8997 3 6.68964 8 3.334 5.59958 8 3 5 6.9837 3 4.8776 Fig.. Two-dimnsional viw for p =, N h = 48: σ = (lft), σ = 3 (cntr), σ = 55 (right). Fig. 3. Thr-dimnsional viw for p =, N h = 48: σ = (lft), σ = 3 (cntr), σ = 55 (right).
Fig. 4. Two-dimnsional viw for p =, N h = 8: σ = (lft), σ = 4.5 (cntr), σ = (right). Fig. 5. Thr-dimnsional viw for p =, N h = 8: σ = (lft), σ = 4.5 (cntr), σ = (right). Fig. 6. Two-dimnsional viw for p =, N h = 5: σ = (lft), σ = 4.5 (cntr), σ = (right). Fig. 7. Thr-dimnsional viw for p =, N h = 5: σ = (lft), σ = 4.5 (cntr), σ = (right). 4.3 Unstructurd D msh W considr an unstructurd coars msh shown in Fig.. This msh contains 9 triangls and 876 triangls aftr uniform rfinmnt. W only prsnt som rsults for th cas of picwis quadratic approximations. As bfor w vary th pnalty paramtrs σ =, 7, 5, 5. Th limiting pnalty valu is
Fig. 8. Two-dimnsional viw for p = 3, N h = 3: σ = (lft), σ = (cntr), σ = 5 (right). Fig. 9. Thr-dimnsional viw for p = 3, N h = 3: σ = (lft), σ = (cntr), σ = 5 (right). Fig.. Two-dimnsional viw for p = 3, N h = 8: σ = (lft), σ = (cntr), σ = 5 (right). Fig.. Thr-dimnsional viw for p = 3, N h = 8: σ = (lft), σ = (cntr), σ = 5 (right). σ = 9.4676. Th solutions on th coars msh ar shown in Fig. 3 and 4 whras th solutions on a rfind msh ar shown in Fig. 5 and 6. W giv th rror in th L and H norms for all cass in Tabl 4.3. W prsnt in Fig. 7 th numrical convrgnc of th SIPG solution for a good pnalty valu (largr than σ ) and a bad pnalty valu (smallr
Y X Fig.. Unstructurd msh with 9 lmnts. Tabl 3 Numrical rrors for two-dimnsional unstructurd msh simulations. N h p σ σ L rror H rror 9 9.4676.63 5.599 9 7.5 9.4676 6.336 66.5934 9 5 9.4676 5.96833 4.999556 876 9.4676 5.5677943 5.847835 876 7.5 9.4676.84393 4.3895847 876 5 9.4676 8.64 3.8834 Fig. 3. Two-dimnsional viw for unstructurd msh and p =, N h = 9: σ = (lft), σ = 7.5 (cntr), σ = 5 (right). Fig. 4. Thr-dimnsional viw for unstructurd msh and p =, N h = 9: σ = (lft), σ = 7.5 (cntr), σ = 5 (right). 3
Fig. 5. Two-dimnsional viw for unstructurd msh and p =, N h = 876: σ = (lft), σ = 7.5 (cntr), σ = 5 (right). Fig. 6. Thr-dimnsional viw for unstructurd msh and p =, N h = 876: σ = (lft), σ = 7.5 (cntr), σ = 5 (right). 9 6 8 4 7 6 ln(h rror) 5 4 ln(l rror) 3 4 5 4.5 4 3.5 3.5.5 ln(h) 6 5 4.5 4 3.5 3.5.5 ln(h) Fig. 7. Numrical convrgnc rats for th cas σ = 3 (dashd lin) and σ = 5 (solid lin): H rrors (lft) and L rrors (right). than σ ). Picwis linar approximation is usd. Th stabl solution convrgs with th xpctd convrgnc rat (O(h ) for th L rror) whras th unstabl solution dos not convrg as th msh siz dcrass. 4.4 Thr-dimnsional Problm W first xplain how to obtain th angl θ T. Th valu cot θ T is th maximum of cot θ E ovr all msh lmnts E. For a givn lmnt E, th angl θ E is th on that yilds th smallst sin θ E,ξ ovr all dgs ξ of th ttrahdron. W now xplain how to obtain θ E,ξ for givn E and ξ. () Comput th quations of th plans corrsponding to th two facs of 4
5 4 3 - - -3.5 Z Z.5 Y.5.5 Y.5 X 5 4 3 - - -3.5 X Fig. 8. Solution on ttrahdral msh: σ = 5 (lft) and σ = 8 (right). E that shar th common dg ξ. i =,, aie,ξ x + bie,ξ y + cie,ξ z + die,ξ =. () Th normal vctors to thos two facs ar i =,, ni = (aie,ξ, bie,ξ, cie,ξ ). (3) Comput cos θe,ξ and sin θe,ξ : cos θe,ξ = n n, sin θe,ξ = ( (cos θe,ξ ) )/. Th msh contains 864 ttrahdral lmnts. Picwis quadratic approximation is usd. In Fig. 8, w show th numrical solution with pnalty valus σ = 5 and σ = 8. Th limiting pnalty valu for ths simulations is σ = 78.75. Th absolut L rror is.57434 for σ = 5 and.36498 for σ = 8. Th absolut H rror is.8399 for σ = 5 and 9.83439 for σ = 8. 5 Conclusions By prsnting lowr bounds of th pnalty paramtr usful for practical computations, this papr rmovs on known disadvantag of th symmtric intrior pnalty mthods, namly th fact that stability of th mthod is obtaind for an unknown larg nough pnalty valu. Evn though w focusd on th lliptic problms, our improvd corcivity and continuity rsults can b applid to th analysis of th SIPG mthod for tim-dpndnt problms. 5
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