The Implementation of Finite Element Method for Poisson Equation

Transcription

1 Th Implmntation of Finit Elmnt Mthod for Poisson Equation Wnqiang Fng Abstract This is my MATH 574 cours projct rport. In this rport, I giv som dtails for implmnting th Finit Elmnt Mthod (FEM) via Matlab and Python with FEniCs. This projct mainly focuss on th Poisson quation with pur homognous and non-homognous Dirichlt boundary, pur Numann boundary condition and Mixd boundary condition on uint squar and unit circl domain. Symmtric and Unsymmtric Nitsch s mthod will b usd to dal with th non-homognous boundary condition. Som of th functions in this projct wr writtn for [4, 5] and som functions ar from Long Chn packag [2][3]. Th Python cod with FEniCs ar larnd from []. Th Modl Problm For simplicity, I considr th following thr typs of boundary conditions:. Pur Dirichlt boundary condition poisson quation: u = f in, u = g D on, 2. Pur Numann boundary condition poisson quation: u = f in, u = g N on, 3. Mixd boundary condition poisson quation: n u = f in, u = g D on Γ D, u = g N on Γ N. n () (2) (3) whr is assumd to b a polygonal domain, f a givn function in L 2 () and g a givn function in H 2 (). Bfor I giv th dtails, I would lik to introduc som usful notations in this rport. Lt T h to b th partition (Figur.) (typically triangulation) of, with picwis constant msh siz h, i.., h K = diam(k), similarly, to b h = diam(), Γ D to b th Dirichlt boundary, Γ N to b th

2 Figur : Th triangulation on unit squar and unit circl domain Numann boundary, and P(K) to b a finit dimnsion smooth function (typically polynomial) on th rgion K. This spac V h ( H0 ()) will b usd to approximat th variabl u. V h := {v L 2 () v K P(K) K T h, v = 0 on Γ D }. (4) 2 Th wak formula and Canonical Galrkin approximation formula for modl problm with 2. Th wak formula and Galrkin approximation for th pur Dirichlt boundary problm Modl Problm. : th wak formula can b writtn as as follows: find u H () with u = g and a(u,φ) := u φdx = f φdx for φ H0, (5) Th Galrkin approximation formula can b rad as follows: find u h V h with u h = g and a(u h,φ h ) := u h φ h dx = f h φ h dx for φ h V h, (6) Modl Problm.2 : th wak formula can b writtn as as follows: find u H () and a(u,φ) := u φdx = f φdx + g N φds for φ V h, (7) Ky words: FEM, Pur Dirichlt boundary condition, Pur Numann boundary condition, Mixd boundary condition, Symmtric and Unsymmtric Nitsch s mthod. Dpartmnt of Mathmatics,Univrsity of Tnnss, Knoxvill, TN, 37909, wfng@math.utk..du 2

3 Th Galrkin approximation formula can b rad as follows: find u h V h with u h = g and a(u h,φ h ) := u h φ h dx = f h φ h dx + (g N ) k φ h ds for φ h V h, (8) Rmark 2.. Th ncssary (and also sufficint in this stting) condition for xistnc of a solution is f dx + g N ds = 0. Th ncssary condition for uniqunss of a solution is udx = 0. W assum f L 2 () and g L 2 ( ) such that f dx + g ds = 0. Thus, thr xists a uniqu solution in th subst of H 2 () consisting of zro-avrag functions. Rmark 2.2. Th systm arising from th pur Numann modl problm is a singular systm. Thr ar thr popular approachs to dal with this singular systm:. Fix on dgr of frdom on th boundary (strongly impos as Dirichlt point); 2. Lagrang multiplir mthod. Changing th non-constrain problm to a constrain systm; 3. Krylov solvrs. AMG for xampl will do a optimal job if you mak sur that th coars solv dosn t mss with th nullspac. Using an itrativ solvr (lik Jacobi) on th coars solv will do th trick. Modl Problm.3 : th wak formula can b writtn as as follows: find u H () and a(u,φ) := u φdx = f φdx + g N φds for φ V h, (9) Γ N Th Galrkin approximation formula can b rad as follows: find u h V h with u h ΓD = g D and a(u h,φ h ) := u h φ h dx = f h φ h dx + (g N ) k φ h ds Γ N for φ h V h, (0) 2.2 Th wak formula for Symmtric and Unsymmtric Nitsch s mthod 2.2. Symmtric Nitsch s mthod. Th wak formula for Modl Problm. can b writtn as as follows: find u H () and a(u,φ) := u φdx u nφds f (φ) = f φdx Γ D Γ D φ nuds + γ h Γ D Γ D φ nuds + γ h Γ D uvds () uvds for φ V h, (2) 3

4 2. Th wak formula for Modl Problm.3 can b writtn as as follows: find u H () and a(u,φ) := u φdx u nφds Γ N Γ D f (φ) = f φdx + g N φds Γ D Γ D φ nuds + γ h Γ D φ nuds + γ h Γ D uvds (3) uvds for φ V h (4), Unsymmtric Nitsch s mthod. Th wak formula for Modl Problm. can b writtn as as follows: find u H () and a(u,φ) := u φdx u nφds + f (φ) = f φdx + Γ D Γ D φ nuds + γ h Γ D Γ D φ nuds + γ h Γ D 2. Th wak formula for Modl Problm.3 can b writtn as as follows: uvds (5) uvds for φ V h, (6) find u H () and a(u,φ) := u φdx u nφds + Γ N Γ D f (φ) = f φdx + g N φds Poisson Solvr 2.3. Data structur Γ D Γ D φ nuds + γ h Γ D φ nuds + γ h Γ D uvds (7) uvds for φ V h (8), Bfor I giv th Poisson solvr, I would lik to introduc th data structur in Matlab. I will us th initial msh (Figur.2) as an xampl to illustrat th concpt of th componnts.. Th basic data structur ( S Tabl ()) is msh which contains msh.nod: Th nod vctor is just th xy-valu of nod. msh.lm: In th lm matrix, th first and th scond column rprsnt th start nodal indics vctor and th nd nodal indics vctor, rspctivly. msh.dirichlt: Th Dirichlt is th Dirichlt boundary condition dgs. msh.numann: Th boundary is th Numann boundary condition dgs. 2. Anothr main data structur is indics (S Tabl (2))which will provid usful indics. indics.nighbor: indics.nighbor(:nt,:3): th indics map of nighbor information of lmnts, whr nighbor(t,i) is th global indx of th lmnt opposit to th i-th vrtx of th t-th lmnt. indics.lm2dg: indics.lm2dg(:nt,:3): th indics map from lmnts to dgs, lm2dg(t,i) is th dg opposit to th i-th vrtx of th t-th lmnt. 4

5 Figur 2: Th initial msh and th uniformly rfinmnt hirarchical msh indics.dg: indics.dg(:ne,:2): all dgs, whr dg(,i) is th global indx of th i-th vrtx of th -th dg, and dg(,) < dg(,2). indics.bdedg: indics.bdedg(:nbd,:2): boundary dgs with positiv oritntation, whr bdedg(,i) is th global indx of th i-th vrtx of th -th dg for i=,2. Th positiv oritntation mans that th intrior of th domain is on th lft moving from bd- Edg(,) to bdedg(,2). Not that this rquirs lm is positiv ordrd, i.., th signd ara of ach triangl is positiv. If not, us lm = fixordr(nod,lm) to fix th ordr. indics.dg2lm: indics.dg2lm(:ne,:4): th indics map from dg to lmnt, whr dg2lm(,:2) ar th global indics of two lmnts sharing th -th dg, and dg2lm(,3:4) ar th local indics of (S Figur. 3 and Tabl. 2). A 3 2 A 3 A 2 Figur 3: Th local indics of vtrics and dgs Poisson solvr Procss Th Canonical Poisson solvr contains thr main stps: Stp Pr-procss stp: In this phas, w should gt th information of th nodal, lmnt and indics. In this projct, I us my own function InitialMsh to gnrat for th squar domain. For th complx domain, you can us Matlab s PDE tool or th packag in [6] to gnrat th 5

6 Tabl : MESH Data structur in two dimnsion Nodal NO. nod(:,) nod(:,2) i Dirichlt(:,) Dirichlt(:,2) Elmnt nighbor lm2dg NO

7 Tabl 2: Indics data structur in two dimnsion dg Edg NO. dg(:,) dg(:,2) Boundary Elmnt dg2lm EdgNO. lm lm 2 local local BdEdg NO. bddg(:,) bddg(:,2)

8 msh. Stp 2 Procss stp: This phas contains four sub-stps as follow: stp 2. Comput th stiffnss matrix and load vctor of th lmnts stp 2.2 Assmbl th global stiffnss matrix and global load vctor stp 2.3 Dal with th boundary condition (Strongly impos th boundary) stp 2.4 Solv th linar systm AU = F Stp 3 post-procss stp: this phas is just to output th solution and giv it visual form. 3 Tmplats 3. Basis functions tmplat Listing : Quadratur points in -D with barycntric coordinats % Basis functions of P 2 phi(:,) = lambda(:,); 3 phi(:,2) = lambda(:,2); 4 phi(:,3) = lambda(:,3); 5 6 % Gradint Basis functions of P 7 Dphip(:,:,) = ; 8 Dphip(:,:,2) = ; 9 Dphip(:,:,3) = ; 0 % Basis functions of P2 2 phi(:,) = lambda(:,). * (2 * lambda(:,) -); 3 phi(:,2) = lambda(:,2). * (2 * lambda(:,2) -); 4 phi(:,3) = lambda(:,3). * (2 * lambda(:,3) -); 5 phi(:,4) = 4 * lambda(:,2). * lambda(:,3); 6 phi(:,5) = 4 * lambda(:,3). * lambda(:,); 7 phi(:,6) = 4 * lambda(:,). * lambda(:,2); 8 9 % Gradint Basis functions of P2 20 Dphip(:,:,) = (4 * lambda(p,) -). * Dlambda(:,:,); 2 Dphip(:,:,2) = (4 * lambda(p,2) -). * Dlambda(:,:,2); 22 Dphip(:,:,3) = (4 * lambda(p,3) -). * Dlambda(:,:,3); 23 Dphip(:,:,4) = 4 * (lambda(p,2) * Dlambda(:,:,3)+lambda(p,3) * Dlambda(:,:,2)); 24 Dphip(:,:,5) = 4 * (lambda(p,3) * Dlambda(:,:,)+lambda(p,) * Dlambda(:,:,3)); 25 Dphip(:,:,6) = 4 * (lambda(p,) * Dlambda(:,:,2)+lambda(p,2) * Dlambda(:,:,)); % Basis functions of P3 29 phi(:,) = 0.5 * (3 * lambda(:,) -). * (3 * lambda(:,) -2). * lambda(:,); 30 phi(:,2) = 0.5 * (3 * lambda(:,2) -). * (3 * lambda(:,2) -2). * lambda(:,2); 3 phi(:,3) = 0.5 * (3 * lambda(:,3) -). * (3 * lambda(:,3) -2). * lambda(:,3); 32 phi(:,4) = 9/2 * lambda(:,3). * lambda(:,2). * (3 * lambda(:,2) -); 33 phi(:,5) = 9/2 * lambda(:,3). * lambda(:,2). * (3 * lambda(:,3) -); 34 phi(:,6) = 9/2 * lambda(:,). * lambda(:,3). * (3 * lambda(:,3) -); 35 phi(:,7) = 9/2 * lambda(:,). * lambda(:,3). * (3 * lambda(:,) -); 36 phi(:,8) = 9/2 * lambda(:,). * lambda(:,2). * (3 * lambda(:,) -); 37 phi(:,9) = 9/2 * lambda(:,). * lambda(:,2). * (3 * lambda(:,2) -); 38 phi(:,0) = 27 * lambda(:,). * lambda(:,2). * lambda(:,3); 8

9 39 40 % Gradint Basis functions of P3 4 Dphip(:,:,) = (27/2 * lambda(p,) * lambda(p,) -9 * lambda(p,)+). * Dlambda (:,:,); 42 Dphip(:,:,2) = (27/2 * lambda(p,2) * lambda(p,2) -9 * lambda(p,2)+). * Dlambda (:,:,2); 43 Dphip(:,:,3) = (27/2 * lambda(p,3) * lambda(p,3) -9 * lambda(p,3)+). * Dlambda (:,:,3); 44 Dphip(:,:,4) = 9/2 * ((3 * lambda(p,2) * lambda(p,2)-lambda(p,2)). * Dlambda(:,:,3) lambda(p,3) * (6 * lambda(p,2) -). * Dlambda(:,:,2)); 46 Dphip(:,:,5) = 9/2 * ((3 * lambda(p,3) * lambda(p,3)-lambda(p,3)). * Dlambda(:,:,2) lambda(p,2) * (6 * lambda(p,3) -). * Dlambda(:,:,3)); 48 Dphip(:,:,6) = 9/2 * ((3 * lambda(p,3) * lambda(p,3)-lambda(p,3)). * Dlambda(:,:,) lambda(p,) * (6 * lambda(p,3) -). * Dlambda(:,:,3)); 50 Dphip(:,:,7) = 9/2 * ((3 * lambda(p,) * lambda(p,)-lambda(p,)). * Dlambda(:,:,3) lambda(p,3) * (6 * lambda(p,) -). * Dlambda(:,:,)); 52 Dphip(:,:,8) = 9/2 * ((3 * lambda(p,) * lambda(p,)-lambda(p,)). * Dlambda(:,:,2) lambda(p,2) * (6 * lambda(p,) -). * Dlambda(:,:,)); 54 Dphip(:,:,9) = 9/2 * ((3 * lambda(p,2) * lambda(p,2)-lambda(p,2)). * Dlambda (:,:,) lambda(p,) * (6 * lambda(p,2) -). * Dlambda(:,:,2)); 56 Dphip(:,:,0) = 27 * (lambda(p,) * lambda(p,2) * Dlambda(:,:,3)+lambda(p,) * lambda(p,3) * Dlambda(:,:,2) lambda(p,3) * lambda(p,2) * Dlambda(:,:,)); 3.2 Quadratur points in -D with barycntric coordinats function [lambda, wight] = quadptsd( numpts) 2 %% QUADPTS quadratur points in -D with Bar. 3 4 if numpts > 8 5 fprintf( No gauss quadratur for this cas ) 6 nd 7 8 switch numpts 9 cas 0 lambda =[ ]; wight = ; 2 3 cas 2 4 lambda =[ ; ]; 6 wight =[ ; ]; 8 9 cas 3 20 lambda =[ ; ; ]; 9

10 23 wight =[ ; ; ]; cas 4 28 lambda =[ ; ; ; ]; 32 wight =[ ; ; ; ]; cas 5 38 lambda =[ ; ; ; ; ]; wight =[ ; ; ; ; ]; cas lambda =[ ; ; ; ; ; ]; wight =[ ; ; ; ; ; ]; cas 7 67 lambda =[ ; ; ; ; ; ; ]; wight =[ ; ; ; ; 0

11 ; ; ]; cas 8 84 lambda =[ ; ; ; ; ; ; ; ]; wight =[ ; ; ; ; ; ; ; ]; 0 nd 3.3 Quadratur points in 2-D with barycntric coordinats Listing 2: Quadratur points in 2-D with barycntric coordinats function [lambda, wight] = quadpts2d( ordr) 2 %% QUADPTS quadratur points in 2-D with barycntric coordinats. 3 % Rfrncs: 4 % 5 % David Dunavant. 6 % High dgr fficint symmtrical Gaussian quadratur ruls for 7 % th triangl. Intrnational journal for numrical mthods in 8 % nginring. 2(6):29--48, if ordr > 6 0 fprintf( No gauss quadratur for this cas ) nd 2 3 switch ordr 4 cas 5 lambda =[ ]; 6 wight = ; 7 8 cas 2 9 lambda =[ ; ; ]; 22 wight =[ ; ; ];

12 25 26 cas 3 27 lambda =[ ; ; ; ]; 3 wight =[ ; ; ; ]; cas 4 37 lambda =[ ; ; ; ; ; ]; 43 wight =[ ; ; ; ; ; ]; cas 5 5 lambda =[ ; ; ; ; ; ; ]; wight =[ ]; cas lambda =[ ; ; ; ; ; ; ; ; ; 2

13 ]; ; ; 8 wight =[ ]; nd 4 Numrical Exprimnts Canonical Finit Elmnt Mthod Tst.. Squar domain In th first tst, w choos th data such that th xact solution of () on th squar domain Ω = [, ] [, ] is givn by u(x, y) = cos(πx) cos(πy) Figur 4: Th canonical FEM approximation on squar domain with pur Dirichlt boundary Th rrors for th FEM approximation (Fig.7) using q =, 2, 3 on unit squar domain and varying h can b found in Tabl (3). 2. Unit circl domain Th rrors for th FEM approximation (Fig.5) using q =, 2, 3 on unit squar domain and varying h can b found in Tabl (4). 3

14 Tabl 3: Errors of th computd solution on squar domain in Tst. q= q=2 q=3 #lm ku uh kl ordr ku uh kh ordr Figur 5: Th canonical FEM approximation on unit circl domain with pur Dirichlt boundary 4

15 Tabl 4: Errors of th computd solution on unit circl domain in Tst. #lm u u h L 2 ordr u u h H ordr q = q = q =

16 4..2 Tst. 2 In th scond tst, w choos th data such that th xact solution of (3) on th squar domain = [,] [,] is givn by whr th mixd boundary condition ar u(x, y) = cos(πx) cos(πy), u(x,y) = g D, x = and x = u(x,y) = g N = 0, y = and y = Th rrors for th FEM approximation using r =,2,3 and varying h can b found in Tabl (5). Tabl 5: Errors of th computd solution on squar domain in Tst 2. #lm u u h L 2 ordr u u h H ordr q = q = q = Symmtric and Unsymmtric Nitsch s mthod 4.2. Tst. In th first tst, w choos th data such that th xact solution of () on th unit domain = [0,] [0,] is givn by u(x) = xy + sin(πx)sin(πy). 6

17 . Symmtric Nitsch s mthod approximation Th rrors for th symmtric Nitsch s mthod approximation (6) using r =,2,3 and varying h can b found in Tabl (6). Figur 6: Th symmtric Nitsch s mthod approximation on squar domain with pur Dirichlt boundary Tabl 6: Errors of th computd solution (symmtric Nitsch s mthod) on squar domain in Tst. #lm u u h L 2 ordr u u h H ordr q = q = Unsymmtric Nitsch s mthod approximation Th rrors for th unsymmtric Nitsch s mthod approximation using r =,2,3 and varying h can b found in Tabl (7) Tst. 2 In th scond tst, w choos th data such that th xact solution of (3) on th unit circl domain is givn by u(x,y) = sin(π(x 2 + y 2 ))cos(π(x y)). 7

18 Figur 7: Th unsymmtric Nitsch s mthod approximation on squar domain with pur Dirichlt boundary Tabl 7: Errors of th computd solution (unsymmtric Nitsch s mthod) on squar domain in Tst. #lm u u h L 2 ordr u u h H ordr q = q = whr u(x,y) = g D, on Γ D, x 0, u(x,y) = g N, on Γ N, x > 0, with g D = xact solution and g N = x (2 pi x cos(pi (x y)) cos(pi (x x + y y)) pi sin(pi (x y)) sin(pi (x x + y y))) + y (2 pi y cos(pi (x y)) cos(pi (x x + y y)) + pi sin(pi (x y)) sin(pi (x x + y y))). Symmtric Nitsch s mthod approximation Th rrors for th symmtric Nitsch s mthod approximation (8) using r =,2,3 and varying h can b found in Tabl (8). 2. Unsymmtric Nitsch s mthod approximation Th rrors for th unsymmtric Nitsch s mthod approximation (Fig.9) using r =,2,3 and varying h can b found in Tabl (9). 8

19 Tabl 8: Errors of th computd solution (symmtric Nitsch s mthod) on unit circl domain in Tst 2. #lm u u h L 2 ordr u u h H ordr q = Tabl 9: Errors of th computd solution (unsymmtric Nitsch s mthod) on unit circl domain in Tst 2. #lm u u h L 2 ordr u u h H ordr q =

20 Figur 8: Th symmtric Nitsch s mthod approximation on squar domain with pur Dirichlt boundary Figur 9: Th unsymmtric Nitsch s mthod approximation on squar domain with pur Dirichlt boundary Listing 3: Python dmo cod 2 # This is MATH 574 cours projct. W try to solv th Poisson 3 # Equation by using Classical FEM and FEM with Nitsch tchniqus 4 # for pur homognous Dirichlt BC. 5 # \ Dlta u + cu= f in \ Omga 6 # u = g_d on \ Gamma_D 7 # u = g_n on \ Gamma_N 8 # 9 0 from collctions import namdtupl 2 from math import log as ln 3 from dolfin import * 4 5 st_log_lvl( ERROR) 6 st_log_lvl( WARNING) 7 20

21 8 Rsult = namdtupl( Rsult, [ fun_nam, NT, h, L2, H0, H ]) 9 f = Exprssion( 2 * pi * pi * sin(pi * x[0]) * sin(pi * x[]) ) 20 u_xact = Exprssion( x[0] * x[] + sin(pi * x[0]) * sin(pi * x[]) ) df classical_poisson(n,p): 24 function_nam= Classical FEM 25 Standard formulation with strongly imposd bcs. 26 msh = UnitSquarMsh(N, N) NT=msh. num_clls() 29 h = msh.hmin() 30 # print hmin, h 3 # print circumradius, Circumradius(msh) #plot(msh) 34 V = FunctionSpac(msh, CG, p) 35 u = TrialFunction(V) 36 v = TstFunction(V) a = innr(grad(u), grad(v)) * dx 39 L = innr(f, v) * dx 40 bc = DirichltBC(V, u_xact, DomainBoundary()) 4 42 uh = Function(V) 43 solv(a == L, uh, bc) #plot(uh, titl= numric ) 46 #plot(u_xact, msh=msh, titl= xact ) 47 # intractiv() # Comput norm of rror 50 E = FunctionSpac(msh, DG, 6) 5 uh = intrpolat(uh, E) 52 u = intrpolat(u_xact, E) 53 = uh - u norm_l2 = assmbl(innr(, ) * dx, msh=msh) 56 norm_h0 = assmbl(innr(grad(), grad()) * dx, msh=msh) 57 norm_h = norm_l2 + norm_h norm_l2 = sqrt( norm_l2) 60 norm_h = sqrt( norm_h) 6 norm_h0 = sqrt( norm_h0) rturn Rsult( fun_nam=function_nam,nt=nt,h=h, L2=norm_L2, H=norm_H, H0=norm_H0) df nitsch_poisson(n,p): 67 Classical ( symmtric) Nitsch formulation. 68 function_nam= Symmtric Nitsch Mthod 69 msh = UnitSquarMsh(N, N) 70 NT=msh. num_clls()

22 73 V = FunctionSpac(msh, CG, p) 74 u = TrialFunction(V) 75 v = TstFunction(V) bta_valu = 2 78 bta = Constant( bta_valu) 79 h_e = MinFactEdgLngth(msh)#msh. ufl_cll(). max_fact_dg_lngth 80 n = FactNormal(msh) 8 82 a = innr(grad(u), grad(v)) * dx - innr(dot(grad(u), n), v) * ds -\ 83 innr(u, dot(grad(v), n)) * ds + bta * h_e ** - * innr(u, v) * ds L = innr(f, v) * dx -\ 86 innr(u_xact, dot(grad(v), n)) * ds + bta * h_e ** - * innr(u_xact, v) * ds uh = Function(V) 89 solv(a == L, uh) 90 9 # Sav solution to fil 92 #fil = Fil(" poisson.pvd") 93 #fil << uh 94 if N==64: 95 viz=plot(uh) 96 viz. writ_png( snitch ) 97 # plot(uh, titl= numric ) 98 # plot(u_xact, msh=msh, titl= xact ) 99 # intractiv() 00 0 # Comput norm of rror 02 E = FunctionSpac(msh, DG, 4) 03 uh = intrpolat(uh, E) 04 u = intrpolat(u_xact, E) 05 = uh - u norm_l2 = assmbl(innr(, ) * dx, msh=msh) 08 norm_h0 = assmbl(innr(grad(), grad()) * dx, msh=msh) 09 norm_dg = assmbl(bta * h_e ** - * innr(, ) * ds) 0 norm_h = norm_l2 + norm_h0 + norm_dg 2 norm_l2 = sqrt( norm_l2) 3 norm_h = sqrt( norm_h) 4 norm_h0 = sqrt( norm_h0) 5 6 rturn Rsult( fun_nam=function_nam,nt=nt,h=msh.hmin(), L2=norm_L2, H=norm_H, H0=norm_H0) df nitsch2_poisson(n,p): 20 Unsymmtric Nitsch formulation. 2 function_nam= Unsymmtric Nitch Mthod 22 msh = UnitSquarMsh(N, N) 23 NT=msh. num_clls() V = FunctionSpac(msh, CG, p) 26 u = TrialFunction(V) 22

23 27 v = TstFunction(V) bta_valu = 2 30 bta = Constant( bta_valu) 3 h_e = MinFactEdgLngth(msh)#msh. ufl_cll(). max_fact_dg_lngth 32 n = FactNormal(msh) a = innr(grad(u), grad(v)) * dx - innr(dot(grad(u), n), v) * ds +\ 35 innr(u, dot(grad(v), n)) * ds + bta * h_e ** - * innr(u, v) * ds L = innr(f, v) * dx +\ 38 innr(u_xact, dot(grad(v), n)) * ds + bta * h_e ** - * innr(u_xact, v) * ds uh = Function(V) 4 solv(a == L, uh) # plot(uh, titl= numric ) 44 # plot(u_xact, msh=msh, titl= xact ) 45 # intractiv() 46 if N==64: 47 viz=plot(uh) 48 viz. writ_png( nnitch ) 49 # Comput norm of rror 50 E = FunctionSpac(msh, DG, 4) 5 uh = intrpolat(uh, E) 52 u = intrpolat(u_xact, E) 53 = uh - u norm_l2 = assmbl(innr(, ) * dx, msh=msh) 56 norm_h0 = assmbl(innr(grad(), grad()) * dx, msh=msh) 57 norm_dg = assmbl(bta * h_e ** - * innr(, ) * ds) 58 norm_h = norm_l2 + norm_h0 + norm_dg norm_l2 = sqrt( norm_l2) 6 norm_h = sqrt( norm_h) 62 norm_h0 = sqrt( norm_h0) rturn Rsult( fun_nam=function_nam,nt=nt,h=msh.hmin(), L2=norm_L2, H=norm_H, H0=norm_H0) # mthods = [ classical_poisson, nitsch_poisson, nitsch2_poisson] # print Th Mthod:{:d}, format( mthod 7 for m in [,2]: #[0,,2]: 72 for p in [,2]: 73 mthod = mthods[m] # print "Th Mthod:{}". format( mthod) # norm_typ = H 78 23

24 79 R = mthod(n=4,p=p) 80 print "Th mthod: {0} with dgr of polynomial {:}". format(r. fun_nam,p) 8 print "{0: >6s} {: >6s} {2: >0s} {3: >7s} {4: >6s} {5: >7s}". format(" Elm", "h", "L^2", "rat", "H^","rat") 82 h_ = R.h 83 _ = gtattr(r, H ) 84 L2_= gtattr(r, L2 ) 85 for N in [8, 6, 32, 64,28,256,52]: 86 R = mthod(n,p=p) 87 h = R.h 88 NT = R.NT 89 = gtattr(r, H ) 90 L2 = gtattr(r, L2 ) 9 rat = ln(/_)/ln(h/h_) 92 ratl2 = ln(l2/l2_)/ln(h/h_) 93 # print h rror rat 94 print {0:6d} {h:.3e} {L2:.5E} {ratl2:.4f} {:.5E} {rat:.4f }. format(nt,h=h,l2=l2, ratl2=ratl2, =, rat=rat) Rfrncs [] M. S. Alnæs, J. Hak, R. C. Kirby, H. P. Langtangn, A. Logg, and G. N. Wlls, Th FEniCS Manual, Octobr, 20. [2] L. Chn, AFEM@MATLAB: a matlab packag of adaptiv finit lmnt mthods, Tchniqu Rport, (2006). [3], ifem: an innovativ finit lmnt mthods packag in MATLAB, Tchniqu Rport, (2009). [4] W. Fng, X. H, Y. Lin, and X. Zhang, Immrsd finit lmnt mthod for intrfac problms with algbraic multigrid solvr, Commun. Comput. Phys., 5 (204), pp [5] W. Fng, X. H, and Y. L. X. Zhang, Immrsd finit lmnt mthod for intrfac problms with algbraic multigrid solvr, Commun.Comput.Phys., (To appar). [6] P. Prsson and G. Strang, A simpl msh gnrator in matlab, SIAM Rviw, 46 (2004), p