Solving the Poisson Partial Differential Equation using Spectral Polynomial Methods

Transcription

1 Solving the Poisson Prtil Differentil Eqution using Spectrl Polynomil Methods Seungkeol Choe April 8th, 4 Abstrct In this report, we present spectrl polynomil method for solving the Poisson eqution with Dirichlet nd Neumnn boundry conditions respectively on one dimensionl compct intervl We cn control the number, size of elements, nd order of pproximting polynomils to obtin ccurte solutions with fster convergence thn the clssicl finite element method We present spectrl Fourier method for solving Poisson eqution with the sme boundry conditions on n two dimensionl nnulus domin By using the tensor product between one dimensionl spectrl bses nd Fourier bses, we pply the high-order method to rdius direction nd the fst Fourier trnsform to ngulr direction Computtionl Engineering nd Science Progrm, University of Uth

2 Contents Introduction 3 Poisson Eqution 4 Method of Weighted Residuls 4 Spectrl Polynomil Elements Method on n Intervl 6 Mthemticl Formultions 6 Bsis Functions 6 Spectrl Polynomil Method in n Element 7 3 Globl Assembly/Direct Stiffness Summtion 8 4 Applying Boundry Conditions 9 Experiment Results H/P Convergence Test for One-dimensionl Solution High order Polynomil Solution nd Its Convergence 3 Spectrl Fourier Method on -dimensionl Annulus 4 3 Mthemticl Formultions 4 3 Poisson Eqution in Polr Coordintes nd Bsis Functions 4 3 Formultion of Spectrl Polynomil nd Fourier Methods 4 3 Experiment Results 7 3 H/P Convergence Test for Two-dimensionl Solution 7 3 High order Polynomil Solution nd Its Convergence 8 4 Conclusion

3 Introduction The spectrl method is numericl scheme for pproximting the solution of prtil differentil equtions It hs developed rpidly in the pst three decdes nd hs been pplied to numericl simultion problems in mny fields One reson for its brod nd fst cceptnce is its use of vrious systems of infinitely differentible bsis functions s tril functions By choosing n pproprite orthogonl system bsed on domin of orthogonlity, we cn pply the method to periodic nd non-periodic problems, s well s problems defined on vrious domins, such s compct domin, hlf/ll intervls Another dvntge of the spectrl method is its high ccurcy In prticulr, the spectrl polynomil method llows control of both the resolution of element size nd the order of pproximtion This results in exponentil convergence, which is mrked improvement over clssicl finite difference nd other purely element methods For solving the problem, we discretize the domin nd obtin the following pproximtion of infinite series A function u is represented vi truncted series expnsion s follows: u u N = N û n φ n, () where φ n re the bsis functions In generl the Chebyshev polynomils T n or the Legendre polynomils L n or nother member of the clss of Jcobi polynomils Pn α,β re employed s bsis functions In choosing proper bsis function, we pply the following 3 chrcteristics to the cndidte bsis n= Numericl Efficiency After discretiztion, the choice of bsis ffects complexity of mss mtrix Moreover we need consider the efficiency in solving the system of liner equtions For exmple, using the monomils {x k } N k= result in the mtrix whose non-zero component re the hlf But its inverse mtrix is full nd the cost of inverting is dominnt In contrst, the Legendre polynomil bsis composes digonl mss mtrix nd its inverse mtrix cn be clculted very efficiently Conditioning If mtrix system is ill-conditioned the round-off error in the mtrix system cn led to lrge errors in the solution Furthermore the number itertion required in inverting the mtrix using itertive solver cn be increse by the condition number The condition number of monomils nd Lgrnge polynomil re close to p, for the polynomil order p But Legendre polynomil hs condition number P + This condition lso ffects the degree of liner independence of ech bsis function This project seeks to study the fundmentl theory of spectrl method, problems, solvbility, nd to obtin its constructive procedure to pply the method to vrious ppliction problems By checking solutions obtined vi the spectrl method ginst exct solutions, we cn vlidte the method nd see how much we cn sve the effort on discretistion of domin to chieve the sme degree of ccurcy in comprison to clssicl element methods 3

4 In chpter, I investigte spectrl method solutions for the forwrd Poisson problem with Dirichlet nd Neumnn boundry condition on ech end of one dimensionl intervl domin In chpter 3, I investigte the spectrl polynomil method nd Fourier method for solving the forwrd Poisson problem with Dirichlet nd Neumnn boundry condition on inner nd outer boundry circles respectively Our pproch utilizes spectrl polynomil element nd Fourier method in tensor product form As n conclusion we present the result of numericl solution nd its convergence by h/p dptive control Poisson Eqution Science nd engineering disciplines re generlly interested in systems of continuous quntities nd reltions This project focuses on solutions of the Poisson eqution, which ppers in vrious field such s electrosttics, mgnetics, het flow, elstic membrnes, torsion, nd fluid flow For exmple, electrosttics, the governing eqution ppers s Guss s lw in differentil form [3]: E = 4πρ () which indictes tht the chrge within closed sphericl surfce is relted to the electric field E norml to surfce element where ρ is chrge density Since it is known in electrosttics tht the electric field E is conservtive, E is form of grdient of sclr potentil Φ, E = Φ (3) With these two reltionships, we obtin Poisson eqution In this report, the one-dimensionl Poisson eqution is defined s Φ = 4πρ (4) L(u) d u + f = (5) dx where u nd f re defined on Ω In pointwise viewpoint, the one dimensionl Poisson eqution (5) is written s for ll x in (, b) L(u)(x) d u(x) + f(x) =, (6) dx Method of Weighted Residuls According to the Weierstrss pproximtion theorem, for ny given rel vlued continuous solution u on compct intervl [, b] we cn obtin rel polynomil function p of certin degree such tht p uniformly pproximtes u Although the result of convergence t ech point is within predefined 4

5 error bound, this does not stisfy the requirement tht we need to cquire n ccurte solution on specific sitution By imposing certin restrictions, we cn obtin formultion tht stisfies the requirement To describe this, we set generl liner differentil eqution on Ω L(u) = (7) with pproprite initil nd boundry conditions Under certin restriction, we ssume tht the true solution u(x) cn be pproximted by finite series expnsion of the form: u δ (x) = N dof i= û i Φ i (x), (8) where Φ i (x) re polynomil tril functions nd û i re N dof unknown coefficients We ssume the following: û = G D : Dirichlet Boundry Vlue, (9) Φ () =, Φ Ndof (b) = where, b re the boundry of domin Ω () We then define non-zero residul R by: () R(u δ ) = L(u δ ) () Define set of functions, H (Ω) nd norm H (Ω) on the spce s follows: H (Ω) = {v L d (Ω) : dx v L (Ω)}, (3) [ v H (Ω) = v(x) + d ] Ω dx v(x) dx, v H (Ω) (4) We define n inner-product, over H (Ω) s follows: u, v = u(x) v(x)dx, (5) This method is restricted to test functions, v(x) tht stisfy: Ω v, R = (6) For exmple, in the colloction method, the j th test function is the Dirc delt function which evlutes to colloction point x = x j Then we hve = δ j, R = R(u δ )(x)δ j (x)dx = R(u δ )(x j ) = L(u δ )(x j ) (7) Other possible test functions re exmined in [] Ω 5

6 Spectrl Polynomil Elements Method on n Intervl Mthemticl Formultions We will exmine equtions of the form of (5) nd (6) with the following boundry conditions: u() = G D : Dirichlet Condition (8) d dx u(b) = G N : Neumnn Condition (9) Multiplying eqution (5) by test functions v(x) nd integrting by prts, we obtin: v(x) d dx u(x)dx + d dx v(x) d dx u(x)dx = v(x)f(x)dx =, () [ v(x)f(x)dx + v d ] b dx u, () for u, v being sufficiently smooth We consider solutions to problem (5) where the forcing function f is well defined in the sense tht vf + [vu ] b < Therefore we only consider tril solutions to eqution () which lie in H (Ω) nd stisfy the Dirichlet boundry condition We cn define the tril spce by X = {u H u() = G D } () Similrly, the spce of ll test functions is restricted to the functions tht re homogeneous on ll Dirichlet boundries Tht is: V = {v H v() = } (3) For numericl pproximtion, we select finite subspces X δ ( X ) nd V δ ( V) for which eqution () holds In prticulr, we cn define δ by the choice of two different discretiztion pproches : Element size or polynomil order The formultion for the wek solution () cn be stted s: Find u δ X δ, such tht Bsis Functions d dx vδ (x) d dx uδ (x)dx = The spectrl pproximtion of solution u is generlly represented s u(x) = [ v δ (x)f(x)dx + v δ d ] b dx uδ, v δ V δ (4) N dof i= û i Φ i (x) (5) on [, b] To construct the globl bsis functions {Φ i (x)} N dof i=, ech Φ i is represented by the liner combintion of locl bsis functions φ i on ech element in [, b], we sy the element Ω e 6

7 We define the locl bsis functions φ i on [, ] to be rel vlued function with Jcobi polynomil of α = nd β = {P, i } s follows: φ i (ξ) = ξ ( ξ +ξ, ) ( ) i = +ξ P, i (ξ), i P (6), i = P for ll ξ in [, ] Then on single stndrd element [, ], the pproximtion u(ξ) is represented s P e u(ξ) = û e i φ i (ξ), (7) i= for ξ in [, ] The locl bsis function φ e i on generl element [x, x ] is defined by the chnge of vrible for φ i between two intervls [x, x ] nd [, ] Spectrl Polynomil Method in n Element We pply the bsis representtion (7) to wek formultion () with the sme test function {φ q }, to obtin the following: P e û e p d dx φ p, φ q = d dx p= P e p= u e pφ p, φ q = f, φ q (8) for q =,, P e where P e is the order of polynomil on locl element Ω e, sy [x, x ] Integrting by prts, we cn use the fct tht [ ] d d x dx φ p, φ q = dx φ p(x)φ q (x) Applying (9) to (8), we obtin x d dx φ p, d dx φ q (9) [ d dx u(x)φ q(x) ] x x + P e p= ûe p d dx φ p, d dx φ q = P e p= ûe p d φ dx p, φ q = f, φ q (3) Pe p= ûe p d dx φ p, d dx φ q = f, φ q + [ d dx u(x)φ q(x) ] x x (3) for q =,, P e In mtrix form we obtin the following the system of equtions for locl coefficients nd modes: φ e, φ e,p e û e f e φ e, û e f e = + u (x ) u (x ) (3) φ e P e,p e û e P e fp e e φ e P e, φ e P e,p e û e P e fp e e where φ p,q = d dx φ p, d dx φ q, fq e = f, φ q, p, q =,, P e Note tht φ q (x ) = δ q,, φ q (x ) = δ q,pe the orthogonlity on {φ q } Pe q= nd 7

8 3 Globl Assembly/Direct Stiffness Summtion As seen in eqution (8), we hve the finite element pproximtion u δ in terms of the globl modes Moreover, we cn represent u δ by liner combintion of locl modes φ e p : u δ (x) = N dof i= N el P e û i Φ i (x) = û e pφ e p(ξ), (33) e= p= where in this cse P e is the polynomil order of the expnsion nd φ e p(ξ) is reprmetriztion of locl bsis function generl elements To see how the stiffness mtrix chnges fter globl ssembly, we hve the following reltionship between globl coefficients û i nd locl coefficients û e i : û = û û e P e = û e+ = û r, e =,, N el, for some r N dof, nd (35) û e P e = û r, e = N el, r = N dof (36) When we determine û i, i =,, N dof, this property plys role tht we cn reduce the size of system According to the orthogonlity defined in the derivtive of interior modes {φ e q} Pe q=, the following reltionships hold: φ e q,qû e q = fq e, q =,, P e (37) for every element e To derive reltionship between djcent boundry modes, we set in generl 3 elements, e = [x, x ], e = [x, x ], nd e = [x, x 3 ] of polynomil order P, P, nd P, respectively: (34) φ P,û + φ P,P û P = f P u (x ) (38) φ,û + φ,p û P = f + u (x ) (39) φ P,û + φ P,P û P = f P u (x ) (4) φ,û + φ,p û P = f + u (x ) (4) Adding (38) to (39), (4) to (4) nd evluting the bsis functions, we obtin: 5 û + û P (= u ) 5 û P = f P + f (4) 5 û + û P (= u ) 5 û P = f P + f (43) Utilizing equtions (37),(4), nd (43) we cn generte prt of the globl stiffness mtrix showing the ssembly of two djcent locl element mtrix system s follows: Aû = f is defined by (44) 8

9 φ P,P 5 5 φ, φ P,P 5 5 φ P,P û P û û û P û û = f P f P + f f f P f P + f f 4 Applying Boundry Conditions Now we cn pply the boundry conditions defined in (8) This is done by processing the system (44) bout the boundry points x = nd x = b This lters the globl system s follows: Aû = f + u (b) u () = f + G N u () (45) We denote: A = A, A,Ndof A Ndof, A Ndof,N dof, û = û û Ndof, nd f = f f Ndof (46) Since û is known to be G D, we cn modify (45) to: A, A, A,Ndof A Ndof, A Ndof, A Ndof,N dof û û û Ndof = f f Ndof + G N G D (47) which results in system of equtions tht hs one solution A, A,Ndof A Ndof, A Ndof,N dof û û û Ndof = f f Ndof + G N A, A Ndof, G D (48) We employ liner solver to obtin [û,, û Ndof ] T 9

10 Experiment Results We now consider eqution (5) in the viewpoint of convergence hving solution u H k (Ω) = {u j k dj L (Ω)} dx j Assuming discretiztion on uniform domin of equi-spced subintervls of size h, the generl error estimte in the norm H k (Ω) for the h-nd p-type extension process cn be written s []: ɛ H k (Ω) CH µ P (k ) u H k (Ω), (49) where ɛ = u u δ, µ = min(k, P + ), nd C is independent of h, P nd u, but depends on k This mens if solution u lies in H k (Ω) for sufficiently lrge k > P +, then this error estimte shows tht we cn chieve exponentil convergence s we increse the polynomil order P Also in prticulr to h-extension process, the error respect to norm H (Ω) stisfies: ɛ H (Ω) K Ch (5) From [], we see tht the slope of the h-type extension process is relted to the minimum of P + nd the smoothness k of the solution Becuse our experiment involves smooth solutions, we observe the slope of h-type extension grph of errors to be very close to P + 5 Approximtion by Spectrl Method Anlytic Solution 5 537e Figure : Numericl nd exct solution of eqution (5) with polynomil order P = 5 H/P Convergence Test for One-dimensionl Solution In this section we present the result of convergence in both h refinement nd p refinement with the following stedy-stte Poisson differentil eqution: d u(x) = sin(πx), (5) dx for ll x in [, ] with zero Dirichlet nd Neumnn boundry conditions For comprison, the numericl nd exct solutions re depicted in figure () Convergence of h-type extension for eqution (5)

11 5 6 Order 3 Order 4 Order Elements Elements Discrete L Error 9 Discrete L Error Element Size (h) Order Per Element Figure : (Left) Convergence with respect to discrete L norm s function of size of elements This test is performed using the h-type extension with polynomils of order 3, 4, nd 5 respectively Error on the Log-Log xis demonstrtes the lgebric convergence of the h-type extension (Right) Convergence wrt L norm s function of size of polynomil order in semi-log plot This shows the exponentil convergence of p-type extension for smooth solution The tests were performed for p-type extension with element length nd This test seeks to estblish the reltion between size of element nd the ccurcy of pproximtion Utilizing equi-distnce elements, we investigte error As shown in Figure (), s elements decrese in size, the ccurcy of the solution improves As we see the reltionship (5) in the theory, the slope of convergence grph should exhibit slopes of slope 4, 5, nd 6 for the polynomil orders 3, 4, nd 5, respectively The exct outcome is shown in left tble of Tble () Convergence of p-type extension for eqution (5) Since the exct solution is n infinite sum of polynomil function, finite order interpolting tril functions cnnot rech idel convergence It is lso pprnt tht the convergence will stgnte before the error reches mchine precision, shown in right figure of Figure (), the experimented results support the behvior described in eqution (49) High order Polynomil Solution nd Its Convergence In this section we construct polynomil P n of order n defined on [, ], which stisfies the following P n () =, P n () = (5) d k dx k P n() =, d k dx k P n () = (53)

12 Tble : This tble shows the convergence of h-type (left) nd p-type (right) resolution control done bove Figure () Observe tht the slopes of ech order P is P + Polynomil order Error(L ) Slope Error 3 6e 44 Element Size (L ) e e e e 6 4 Plot of nnlytic solution 8 imge domin Figure 3: Exmple of curve stisfing conditions (5), with polynomil order n = 9 for ll k =,, n For ech n, we obtin polynomil P n by solving system of liner equtions tht determines the coefficients of P n We pply the spectrl polynomil solver to pproximte the second derivtive Q n of P n The numericl nd exct solutions by the solver we developed is shown in figure (3) Problem Consider the following differentil eqution for u(x) such tht d dx u(x) = Q n, (54) for ll x in [, ] with the boundry condition defined in eqution (5) spectrl polynomil method Approximte u(x) using Note tht the ccurcy of the interpoltion stisfying equtions (5) is dependent on the stbility of the mtrix defining the coefficients of interpolnts We used the Legendre bsis functions becuse they re known to be more stble thn monomils Despite this choise, interpoltion error is nerly e 3 This results in the sme mount of convergence error in p-type extension mode shown in right of Figure (4) nd Tble () Convergence h-type extension for eqution (54) Exmining the eqution (5), in Figure (4), we observe tht the error with respect to L of the discrete solution to the eqution is exponentilly convergent with respect to the size of element This verifies the Log-Log scle of reltion of theory (5)

13 Convergence p-type extension for eqution (54) This semi-log scle plot lso shows the exponentil convergence of p-type extension of tril functions Note tht we pproximte the finite order of the polynomils So there exists the lowest order P l tht pproximtes with tril functions of order P which P > P l should shows the sme convergence s the cse using tril functions of order P l In right of Figure (4), we observe tht the convergence is stying on pproximtion error which theoreticlly should be mchine precision 4 5 Order 3 Order 4 Order 5 5 Elements Elements Discrete L Error 8 9 Discrete L Error Element Size (h) Order Per Element Figure 4: (Left) Convergence with respect to discrete L norm s function of element size This test is performed using the h-type extension with fixed polynomil order 3, 4, nd 5 respectively Error on the Log-Log xis demonstrtes the lgebric convergence of the h-type extension (Right) Convergence wrt L norm s function of size of polynomil order in semi-log plot It shows the exponentil convergence of p-type extension for smooth solution The two tests re performed for p-type extension with element lengths of nd Tble : This tble shows the convergence of h-type resolution control done bove Figure (4) Note tht the slopes of ech order P is P + Polynomil order Error(L ) Slope e e e Error Element Size (L ) 343e 3 386e 3 3

14 3 Spectrl Fourier Method on -dimensionl Annulus 3 Mthemticl Formultions 3 Poisson Eqution in Polr Coordintes nd Bsis Functions We formulte the Generlized Poisson problem on n nnulus [, b] [, π], > under the periodic solution u s follows: [ r (σ(r, θ) r ) + r (σ(r, θ) r ) + r θ (σ(r, θ) ] θ ) u(r, θ) = f(r, θ), (55) where r [, b] nd θ [, π] The boundry conditions for this domin is given by: where θ [, π] u(, θ) = G D (θ), with periodicity of u, u(r, ) = u(r, π), (56) r u(b, θ) = G N(θ), (57) The representtion of pproximtion of u is gurnteed by Weierstrss theorem: u(r, θ) = N r N θ / j= k= N θ /+ û jk φ j (r)e ikθ, (58) where r [, b] nd θ [, π] for the globl degree of freedom N r nd N θ on û jk s The bsis functions {φ j } Nr j= shown t liner spn (58) re defined s modified Jcobi polynomils defined in [] As review of discrete Fourier trnsform in N-point grid described in [], the formul for the discrete Fourier trnsform for {v j } is where x j = j π N N ˆv k = h e ikx j v j, k = N +,, N, (59) j= nd the inverse discrete Fourier trnsform for {ˆv k} is given by v j = π Nr/ k= N/+ e ikxj ˆv k, j =,, N (6) 3 Formultion of Spectrl Polynomil nd Fourier Methods In this project, we ssume the conductivity term σ in eqution (55) to be only dependent on vribles showing the rdius domin s we multiply r in ech side of eqution (55) Then the Poisson eqution in polr coordinte is s follows: ] [r (σ(r) ) + rσ(r) + σ(r) r r r θ u(r, θ) = r f(r, θ) (6) 4

15 We utilize the Glerkin method, with test functions of the form: φ p (r)e iqθ, p =,, N r, q = N θ +,, N θ (6) The wek form of the eqution is s follows: r r (σ r u) + rσ r u + σ θ u, φ pe iqθ = r f, φ p e iqθ (63) Define T i, i =,, 4 s follows T = T = T 3 = nd T 4 = Then eqution (63) becomes: π π π π φ p e iqθ r r [σ(r) r u(r, θ) ] drdθ, (64) φ p e iqθ rσ(r) u(r, θ)drdθ, (65) r φ p e iqθ σ(r) u(r, θ)drdθ, (66) θ φ p e iqθ r f(r, θ)drdθ (67) T T T 3 = T 4 (68) We obtin boundry terms by integrting by prts on T s follows: T = π e [r iqθ σ(r) ] b r u(r, θ)φ p(r) π π Then the right hnd side of (68) becomes π T T T 3 = + + dθ (69) e iqθ rσ(r) r u(r, θ)φ p(r)drdθ (7) e iqθ r σ(r) r u(r, θ) d dr φ p(r)drdθ (7) π π π e [r iqθ σ(r) ] b r u(r, θ)φ p(r) dθ (7) e iqθ rσ(r) r u(r, θ)φ p(r)drdθ (73) e iqθ r σ(r) r u(r, θ) d dr φ p(r)drdθ (74) e iqθ σ(r) θ u(r, θ)φ p(r)drdθ (75) Using (58) nd the orthogonl properties of {e ikθ }, k = N θ +,, N θ, we cn simplify (7) to obtin: 5

16 π T T T 3 = e [r iqθ σ(r) ] b r u(r, θ)φ p(r) dθ (76) N θ + π û jq rσ(r) d dr φ j(r)φ p (r)dr (77) Let us define the following mtrices: j= N θ + π û jq r σ(r) d dr φ j(r) d dr φ p(r)dr (78) j= N θ + π û jq q σ(r)φ j (r)φ p (r)dr (79) j= where j, p =,, N θ T 4 becomes: (M ) jp = (M ) jp = (M 3 ) jp = rσ(r) d dr φ j(r)φ p (r) dr (8) r σ(r) d dr φ j(r) d dr φ p(r) dr (8) σ(r)φ j (r)φ p (r) dr, (8) T 4 = = π r φ p (r) φ p (r)e iqθ r f(r, θ)dθdr (83) π f(r, θ)e iqθ dθdr (84) (85) For given r, the Discrete Fourier Trnsform for f(r, θ) is defined by where f(r, θ τ ) = π f(r) k = π N θ N θ / k= N θ /+ N θ e ikθτ f(r) k (86) e ikθ j f(r, θ j ) (87) j= with k { N θ +,, N θ } nd θ j { π N θ,, π} Then: 6

17 T 4 = = π r φ p (r) π N θ / k= N θ /+ e ikθ f(r)k e iqθ dθdr (88) r φ p (r) f(r) q dr (89) N θ π û jq M jp + π û jq M jp + π û jq q M 3jp = j= +b σ(b)φ p (b) N θ j= π N θ j= e iqθ G N (θ)dθ σ()φ p () π r φ p (r) f(r) q dr (9) e iqθ u(, θ)dθ (9) r where j, p =,, N r We pply the Fourier trnsform to the integrl term with G N nd the sme ide s one-dimensionl cse to ech term bout boundry conditions 3 Experiment Results 3 H/P Convergence Test for Two-dimensionl Solution In this section we present the convergence behvior in both h refinement nd p refinement for the following stedy-stte Poisson differentil eqution: r r (σ r u) rσ r u σ θ u = r cos θ [ 4rπ cos S r + {4π + } sin S r ], (9) for ll r [, ], θ [, π] with σ(r) = r The nlytic solution is known to be u(r, θ) = sin S r cos θ, (93) where S r = π(r ) π The numericl tht hs sme shpe s exct solutions re shown in figure (5) with different viewpoint Tble 3: This tble shows the convergence of h-type (left) nd p-type (right) resolution control done bove Figure (6) We cn see the slope of ech cse is P + of order P Polynomil order Error(L ) Slope Element Size Error(L ) e e e 346e 3 7

18 5 Approximtion Vlue 5 Approximtion Vlue θ 4 Rdius θ Rdius Figure 5: Numericl solution of eqution (9) with polynomil order P =, equidistnce elements The difference to exct solution is 4774e-5 in terms of L norm 3 High order Polynomil Solution nd Its Convergence In this section we construct polynomil P n of order n defined on [, ], which stisfies the following P n () =, P n () = (94) d k dx k P n() =, d k dx k P n () = (95) for ll k =,, n For ech n, we obtin polynomil P n by solving system of liner equtions tht determines the set of coefficients of P n We use the spectrl polynomil solver to pproximte the second derivtive Q n of P n We investigte the convergences by the h-type, p-type extension of tril functions below Problem 3 Consider the following differentil eqution for u(x) such tht r r (σ r u) rσ r u σ θ u = e (r ) {P n (r) + (r (r ) r)p n(r) r P n (r)} cos θ, (96) for ll r in [, ] with σ(r) = e (r ) The exct solution is known to be: u(r, θ) = P n (r) cos θ (97) where r in [, ] nd θ in [, π] 8

19 6 Order 5 Order 6 5 Elements Elements Discrete L Error 9 Discrete L Error Element Size (h) Order Per Element Figure 6: (Left) Convergence with respect to discrete L norm s function of size of elements This test is performed using the h-type extension with fixed polynomil order 5 nd 6 respectively Error on the Log-Log xis is demonstrting the lgebric convergence of the h-type extension (Right) Convergence wrt L norm s function of size of polynomil order in semi-log plot It shows the exponentil convergence of p-type extension for smooth solution Two tests re performed for p-type extension with element length nd Tble 4: This tble shows the convergence of h-type resolution control done bove Figure (8) We cn see the slopes of ech order P is P + Polynomil order Error(L ) Slope Element Size Error(L ) e e e e Exct Solution 5 5 Approximtion Vlue θ 4 6 Rdius θ 4 6 Rdius 8 Figure 7: Exmple of curve tht stisfies conditions (94) with polynomil order n = 7, 9,, 4, 5 The error is 939e 8 9

20 5 Order 5 Order 6 5 Elements Elements 6 4 Discrete L Error 7 8 Discrete L Error Element Size (h) Order Per Element Figure 8: (Left) Convergence with respect to discrete L norm s function of size of elements This test is performed using the h-type extension with fixed polynomils of order 3, 4, nd 5 respectively Error on the Log-Log xis demonstrtes the lgebric convergence of the h-type extension (Right) Convergence wrt L norm s function of size of polynomil order in semi-log plot It shows the exponentil convergence of p-type extension for smooth solutions The two tests re performed for p-type extension with element lengths of nd

21 4 Conclusion Throughout this project, we investigte the ppliction of Spectrl Polynomil Element Method to Poisson equtions We lso compred the of h/p convergence properties of the method to the clssicl finite element method The Glerkin method llows incorportion of the wek solution into the formultion for the problem s system of liner equtions which cn be solved numericlly The Spectrl element solver for one dimensionl Poisson eqution with Dirichlet nd Neumnn boundry conditions, with high-order solutions exhibited much ccurte solutions which were hrd to get cceptble convergence in given time nd resolution of domin in pst For the future study, we would like del with problems regrding: development multi-dimensionl solver obtining solutions to vrious nturl phenomen tht obey governing equtions utilizing the method in the ill-posed problem By using certin technique tht we cn pproximte the solution, we cn lso pply this method to the problem nd compre with other method in tht sitution References [] Spectrl/Hp Element Methods for Cfd George Em Krnidkis, Spencer J Sherwin, Oxford Univ Press, 999 [] Spectrl Methods in MATLAB Lloyd N Trefethen, Society for Industril nd Applied Mthemtics, [3] Lecture note of Advnced Methods in Scientific Computing Christopher R Johnson, School of Computing, University of Uth, [4] A direct spectrl colloction Poisson solver in polr nd cylindricl coordintes, Chen HL Su YH nd Shizgl BD, Journl of Computtionl Physics 6(), ()