Hybridizable discontinuous Galerkin (HDG) method for Oseen flow

Transcription

1 Hybrdzable dscontnuous Galerkn (HDG) method for Oseen flow Matteo Gacomn Laborator de Càlcul Numèrc, E.T.S. de Ingeneros de Camnos, Unverstat Poltècnca de Catalunya BarcelonaTech, Jord Grona 1, E Barcelona, Span July 13, 2017 Contents 1 Introducton 2 2 Problem statement 2 3 The hybrdzable dscontnuous Galerkn (HDG) formulaton HDG local problems HDG global problem Local post-process of the velocty feld Assembly of the matrces Exercse: the Kovasznay flow [oseen/ex 2] 8 A Blnear and lnear forms 8 Emal address: matteo.gacomn@upc.edu 1

2 1 Introducton Ths short note complements the computer lab on hybrdzable dscontnuous Galerkn for the approxmaton of the Oseen flow and s the natural follow-up of the prevous sessons. The assocated academc Matlab code solves problems n two dmensons usng meshes based on trangular elements. Ths lab sesson s devoted to the expermental analyss of the role of the stablzaton parameter n problems featurng advecton phenomena. For a general analyss of HDG for Oseen flow, the nterested reader s referred to [1]. 2 Problem statement Let Ω R nsd be an open bounded doman wth boundary Ω. The strong form of the steady Oseen equaton wth non-homogeneous Drchlet boundary condtons reads as follows: (K u u a pi nsd ) = s n Ω, u = 0 n Ω, (1) u = u D on Ω, where the couple (u, p) represents the velocty and pressure felds assocated wth the problem, K = νi nsd, ν > 0 s the vscosty matrx, a a dvergence-free advecton feld and s and u D respectvely are the volumetrc source term and the Drchlet boundary datum to mpose the value of the velocty on Ω. Remark that owng to the purely Drchlet boundary condton, the pressure n (1) s determned up to a constant. Hence, the followng addtonal constrant enforcng zero mean value of the pressure feld s ntroduced: p dγ = 0. (2) Assume that Ω s parttoned n n el dsjont subdomans Ω s Ω n el n el Ω = Ω, Ω Ω j = for j, Ω h := Ω, =1 whose boundares Ω defne the nternal nterface Γ Γ := [ n el =1 ] Ω \ Ω. (3) =1 Moreover, the jump operator along each porton of the nterface Γ s defned as the sum of the values of the quantty under analyss n the elements Ω and Ω j respectvely on the left and rght sdes of the nterface (cf. [2]), that s, = + j. 2

3 It s mportant to observe that ths defnton always requres the normal vector n n the argument and produces functons n the same space as the argument. The orgnal strong problem (1) can thus be rewrtten equvalently n mxed form on the broken doman as follows: L u = 0 n Ω, and for = 1,..., n el, (K L u a pi nsd ) = s n Ω, and for = 1,..., n el, u = u D on Ω, u n = 0 on Γ, ( K L ) n + ( u a ) n + pn = 0 on Γ. u = 0 n Ω, and for = 1,..., n el, where last two equatons - also known as transmsson condtons - enforce the contnuty of respectvely the prmal varable and the normal trace of the flux across the nterface Γ. In order for pressure to be unquely defned, (4) s coupled agan wth the constrant (2). (4) 3 The hybrdzable dscontnuous Galerkn (HDG) formulaton The hybrdzable dscontnuous Galerkn method was frst ntroduced n [3]. In ths secton, the method s used to approxmate the Oseen equaton [1]. Frst, followng the notaton n [4], the dscrete functonal spaces are ntroduced: V h (Ω) := {v L 2 (Ω) : v Ω P k (Ω ) Ω, = 1,..., n el }, M h (S) := {ˆv L 2 (S) : ˆv Γ P k (Γ ) Γ S Γ Ω}, where P k (Ω ) and P k (Γ ) stand for the spaces of polynomal functons of complete degree at most k respectvely n Ω and on Γ. Moreover, recall the notaton (p, q) V := pq dω V and p, q S := pq dγ for the classcal nternal products n L S 2(V ), V Ω and L 2 (S), S Γ. The hybrdzable dscontnuous Galerkn formulaton of the Oseen equaton ntroduced n secton 2 s composed by n el local problems defned on the nteror of the elements Ω s and a global problem set on the nternal skeleton Γ. In the followng subsectons, the aforementoned problems wll be detaled. 3.1 HDG local problems The local problems determne (u, L, p ) for each element Ω, = 1,... n el as functons of a new varable û defned along the nterface Γ and actng as a Drchlet boundary condton, 3

4 namely L u = 0 n Ω, (K ) L u a p I nsd = s n Ω, u = 0 n Ω, u = u D u = û on Ω Ω, on Ω \ Ω, Henceforth, assume that the hybrd varable û s defned on both the nternal skeleton Γ and the external boundary Ω. More precsely, let û = u D on Ω. It follows that the boundary condtons n (5) may be rewrtten smply as (5) u = û on Ω. (6) By multplyng the frst three equatons n (5) by test functons belongng to approprate functonal spaces and ntegratng by parts, the followng weak form of the local problems s derved. For = 1,... n el, seek (u h, L h, p h ) [V h (Ω )] nsd [V h (Ω )] nsd nsd V h (Ω ) such that for all (w, G, q) [V h (Ω )] nsd [V h (Ω )] nsd nsd V h (Ω ) t holds (G, L h ) Ω + ( G, u h ) Ω = Gn, û h Ω, (7a) (w, (K L h )) Ω + (w, L h a) Ω +(w, p h ) Ω + w, τ u h Ω = (w, s) Ω + w, τ û h Ω, (7b) ( q, u h ) Ω = q, û h n Ω, 1, p h Ω = ρ, where the trace of the numercal flux s defned element-by-element as ( K L h + u h a + ph I ) n sd n = ( ) K L h n + ( u h a ) n + p h n + τ (u h û h ). (8) Remark that the momentum equaton (7b) has been ntegrated twce n order to make the flux assocated wth the operator (frst three terms on the rght-hand sde of (8)) vansh by leavng solely the stablzaton term w, τ (u h û h ) Ω. Moreover, n order to account the purely Drchlet boundary condtons (6) appled on the local problem, the constrant (7d) has been ntroduced. The stablzaton parameter τ plays a crucal role n the accuracy and convergence of the HDG method (cf. e.g. [1, 5, 6]) and may assume dfferent values on each face of the boundary Ω. Its role n the HDG approxmaton of the Oseen flow s the man subject of ths lab sesson and wll be nvestgated va expermental analyss. (7c) (7d) 3.2 HDG global problem As prevously mentoned, the hybrd varable û h ntroduced n (7) s the unknown of a global problem whch accounts for the transmsson condtons n (4): seek û h [M h (Γ 4

5 Ω)] nsd such that û h = u D on Ω and for all ˆv [M h (Γ Ω)] nsd t holds n el =1 { ˆv, ( ) K L h n Ω \ Ω + ˆv, ( û h a ) n Ω Γ + ˆv, p h n Ω \ Ω, (9a) } + ˆv, τ u h Ω \ Ω ˆv, τ û h Ω \ Ω = 0 1, û h n Ω = 0 for = 1,..., n el (9b) The local problems (7) featurng purely Drchlet boundary condtons, pressure s known up to a constant. In order to mpose an addtonal constrant, equaton (7d) sets the mean value of p h on the boundary of all elements Ω s, for = 1,..., n el. Thus, the compatblty condton (9b) to weakly enforce a dvergence-free velocty feld on each element s ntroduced. 3.3 Local post-process of the velocty feld By solvng an addtonal problem element-by-element, a post-processed velocty feld may be computed. More precsely, when usng an approxmaton based on polynomals of order k, the optmal order of convergence k + 1 of the gradent of the velocty L s exploted to obtan a velocty feld u superconvenvergng wth order k + 2. The dea of post-processng the velocty feld was frst proposed n [7] and stems from the BDM-projecton used n mxed methods [8, 9] to obtan optmal order of convergence for the flux. The post-processed velocty s obtaned by solvng the followng problem n each element: { ( u ) = L h n Ω, ( ) (10) u n = L h n on Ω, wth the addtonal constrant u dω = Ω u h dω Ω for = 1,..., n el. (11) Henceforth, to smplfy the notaton the superndex h expressng the dscrete approxmatons and the subndex ndcatng the element wll be dropped, unless needed n order to follow the development. 3.4 Assembly of the matrces Consder an element-by-element nodal nterpolaton for the spatal unknown functons defned as û(x) û h (x) = n fn j=1 N j (x) û j [M h (Γ Ω)] nsd, (12a) 5

6 n en u(x) u h (x) = N j (x) u j [V h (Ω h )] nsd, (12b) j=1 n en L(x) L h (x) = N j (x) L j [V h (Ω h )] nsd nsd, (12c) j=1 n en p(x) p h (x) = N j (x) p j V h (Ω h ), (12d) j=1 (12e) where û j, u j, L j and p j are nodal values, N j are polynomal shape functons of order k n each element, n en s the number of nodes per element, N j are the polynomal shape functons of order k on each element face/edge, and n fn s the correspondng number of nodes per face/edge. Hence, the vectors u, L and p are defned for each element = 1,..., n el and are respectvely of dmenson n en n sd, n en n 2 sd and n en. The vector û s defned globally over the skeleton of the mesh and ts dmenson corresponds to the number of nodes on Γ Ω. More precsely, n ef dm(û) = n k fn, where n ef s the number of element faces/edges of the mesh skeleton Γ Ω and n k fn s the number of nodes on the k-th face. Remark that owng to the boundary condton û h = u D on Ω, solely the nodes belongng to Γ are unknowns of the prevously ntroduced global problem. k=1 Algebrac system arsng from the local problem The matrx formulaton of the local problem (7) for each element Ω for = 1,..., n el reads as follows: A uu A ul A up 0 u f u A Lu A LL 0 0 L f A pu 0 0 A T = L + ρp p f p 0 0 A ρp 0 λ 0 A uû A Lû A pû û + ρ 0 (13) 1 where a Lagrange multpler λ has been ntroduced to handle the addtonal constrant to enforce unqueness of the pressure feld and the matrx A ρp arses from the left-hand sde of equaton (7d). Remark that the varable ρ features solely one degree of freedom per element. The local block matrces A are computed explotng the nterpolaton framework dscussed above and the correspondng forms a n appendx A. 6

7 Algebrac system arsng from the global problem Smlarly, the followng algebrac formulaton may be derved for the global problem (9): n el { ] u [Aûu AûL Aûp L + [ } Aûû ] û = 0 (14) =1 p where the block matrces are computed as mentoned above startng from the forms dscussed n appendx A. By computng the nverse of the matrx n (13), the soluton {u, L, p, λ} T may be wrtten as a functon of û. By ntroducng a restrcton operator R that neglects the last lne assocated wth the Lagrange multpler and by pluggng the resultng vector {u, L, p} T nto (14), the followng global problem s derved where [ ] Âûû = A nel =1 Aûu AûL Aûp [ ] Âûρ = A nel =1 Aûu AûL Aûp R R [ ] fû = A nel =1 Aûu AûL Aûp Âûû û + Âûρρ = fû (15) 1 A uu A ul A up 0 A uû A Lu A LL 0 0 A Lû A pu 0 0 A T ρp A pû + [ Aûû ], 0 0 A ρp A uu A ul A up 0 0 A Lu A LL A pu 0 0 A T ρp 0, 0 0 A ρp A uu A ul A up 0 f u A Lu A LL 0 0 f L A pu 0 0 A T ρp f p. 0 0 A ρp 0 0 R Accountng for the global purely Drchlet boundary condton The compatblty condton (9b) may be rewrtten as  ρû û = 0, (16) where Âρû s the matrx assocated wth the blnear form on the left-hand sde of (9b). The fnal algebrac global system s obtaned from (15) and (16): ] {û } } [Âûû Âûρ { fû =, (17)  ρû 0 ρ 0 whose unknowns are the the trace of the velocty û defned on the mesh skeleton Γ Ω and the mean boundary pressures ρ defned n all the elements. 7

8 4 Exercse: the Kovasznay flow [oseen/ex 2] Let Ω R 2 be the square doman [ 1, 1] [ 1, 1] and consder the analytcal soluton of the ncompressble Naver-Stokes equatons known as Kovasznay flow [10]: ( ( u 1 = 1 e λ(x+1) cos 2π y + 1 )), 2 u 2 = λ ( ( 2π eλ(x+1) sn 2π y + 1 )), 2 p = 1 2 e2λ(x+1) + C, where λ := 1/(2ν) 1/(4ν 2 ) + 4π 2 and C s a constant such that the exact pressure has zero mean value on the boundary Ω. By settng the advecton feld a equal to the analytcal soluton u and by computng approprately the Drchlet boundary datum u D and the source term s, the Kovasznay flow s also a soluton of the Oseen equaton. Dffuson-domnated and advecton-domnated regmes Solve the Oseen equaton for ν = 10 1 and ν = 10 4 by consderng a stablzaton term constant on each face Γ of the trangulaton. More precsely, set τ = τ (1) := ν/l, l beng a characterstc length of the doman. Repeat the experments by settng τ = 10 κ ν/l, wth κ = 1,..., 3. Incorporatng the advecton nformaton nto the stablzaton term Implement the followng two expressons of the stablzaton parameter constant on each face Γ : τ (2) := ν l max a(x) n, (3) τ := ν x Γ l max { } a(x) n, 0 x Γ Optmal order of convergence and superconvergence of the post-processed velocty Recall that for a polynomal approxmaton of order k, the optmal order of convergence for velocty s k +1 and the one for the post-processed velocty s k +2. Verfy expermentally the order of convergence of the velocty for ν = 10 1 and ν = 10 4 for dfferent meshes mesh2 P,...,mesh5 P and for dfferent order of approxmaton degree = 1,..., 5. Compare the results obtaned usng dfferent expressons of the stablzaton parameter. A Blnear and lnear forms Forms used n the HDG global problem (14): aûu (ˆv, u):= ˆv, τ u Ω \ Ω, aûp (ˆv, p):= ˆv, p n Ω \ Ω, aûl (ˆv, L):= ˆv, ( KL ) n Ω \ Ω, aûû (ˆv, û):= ˆv, ( û a ) n Ω Γ ˆv, τ û Ω \ Ω. 8

9 Forms used n the HDG local problem (13): a uu (w, u):= w, τ u Ω, a ul (w, L):= ( w, (KL )) a up (w, p):= ( w, p )Ω a uû (w, û):= w, τ û Ω, a LL (G, L):= ( G, L ) Ω, a Lu (G, u):= ( G, u )Ω, a Lû (G, û):= Gn, û Ω, a pu (q, u):= ( q, u )Ω, a pû (q, û):= q, û n Ω, f u (w):= ( w, s ) Ω. Ω + ( w, L a ) Ω References [1] A. Cesmeloglu, B. Cockburn, N. C. Nguyen, and J. Perare, Analyss of HDG methods for Oseen equatons, J. Sc. Comput., vol. 55, no. 2, pp , [2] A. Montlaur, S. Fernández-Méndez, and A. Huerta, Dscontnuous Galerkn methods for the Stokes equatons usng dvergence-free approxmatons, Int. J. Numer. Methods Fluds, vol. 57, no. 9, pp , [3] B. Cockburn, J. Gopalakrshnan, and R. Lazarov, Unfed hybrdzaton of dscontnuous Galerkn, mxed, and contnuous Galerkn methods for second order ellptc problems, SIAM J. Numer. Anal., vol. 47, no. 2, pp , [4] R. Sevlla and A. Huerta, Tutoral on Hybrdzable Dscontnuous Galerkn (HDG) for second-order ellptc problems, n Advanced Fnte Element Technologes (J. Schröder and P. Wrggers, eds.), vol. 566 of CISM Internatonal Centre for Mechancal Scences, pp , Sprnger Internatonal Publshng, [5] B. Cockburn, B. Dong, J. Guzmán, M. Restell, and R. Sacco, A hybrdzable dscontnuous Galerkn method for steady-state convecton-dffuson-reacton problems, SIAM J. Sc. Comput., vol. 31, no. 5, pp , [6] N. C. Nguyen, J. Perare, and B. Cockburn, An mplct hgh-order hybrdzable dscontnuous Galerkn method for lnear convecton-dffuson equatons, J. Comput. Phys., vol. 228, no. 9, pp , [7] B. Cockburn, B. Dong, and J. Guzmán, A superconvergent LDG-hybrdzable Galerkn method for second-order ellptc problems, Math. Comp., vol. 77, no. 264, pp , cted By 92. [8] P.-A. Ravart and J. M. Thomas, A mxed fnte element method for 2nd order ellptc problems, pp Lecture Notes n Math., Vol. 606, [9] D. N. Arnold and F. Brezz, Mxed and nonconformng fnte element methods: mplementaton, postprocessng and error estmates, RAIRO Modél. Math. Anal. Numér., vol. 19, no. 1, pp. 7 32,

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