Spline Element Method for Partial Differential Equations

Transcription

1 for Partial Differential Equations Department of Mathematical Sciences Northern Illinois University 2009 Multivariate Splines Summer School, Summer 2009

2 Outline 1 Why multivariate splines for PDEs? Motivation Features of the method Approximation properties 2 Uniform refinement of a tetrahedron Numerical methods for saddle point problems 3 4 Robustness of the method for singular perturbation problems Fourth order singular problem Darcy-Stokes equation 5 Navier-Stokes equations

3 Motivation Features of the method Approximation properties 1 Why multivariate splines for PDEs? Motivation Features of the method Approximation properties 2 Uniform refinement of a tetrahedron Numerical methods for saddle point problems 3 4 Robustness of the method for singular perturbation problems Fourth order singular problem Darcy-Stokes equation 5 Navier-Stokes equations

4 Motivation Features of the method Approximation properties Finite element implementation with Lagrange multipliers The finite element method is the most widely used method for solving numerically partial differential equations A collection of methods falls under the designation f.e.m. Model problem { u = f in Ω u = g on Ω where Ω will denote the boundary of the bounded domain Ω and denotes the Laplace operator, = n i=1 2. xi 2 Green s identity Ω ( div u)v dx = Ω u u v dx Ω ν v.

5 Motivation Features of the method Approximation properties Find u in H0 1 (Ω) such that u v = Ω Ω fv, v H 1 0 (Ω), Biharmonic equation Ω 2 u = f in Ω u = g on Ω u n = h in Ω, u v dx = Ω f v, v H0 2 (Ω), (1) Abstract variational problem Find u V such that Lax Milgram lemma a(u, v) = l, v for all v V

6 Motivation Features of the method Approximation properties Discrete approximations Find u V h such that a(u, v) = l, v for all v V h Cea s lemma u u h V C min v Vh u v V for a constant C independent of h. Requirements for conforming approximations Let k 1 and suppose Ω is bounded. Then a piecewise infinitely differentiable function v : Ω R belongs to H k (Ω) if and only if v C k 1 (Ω). Even in two dimensions, conforming finite element spaces can be very complicated and there are no satisfactory answer in three dimesnions. Nonconforming approximations are not very popular.

7 Motivation Features of the method Approximation properties Discrete approximations Find u V h such that a(u, v) = l, v for all v V h Cea s lemma u u h V C min v Vh u v V for a constant C independent of h. Requirements for conforming approximations Let k 1 and suppose Ω is bounded. Then a piecewise infinitely differentiable function v : Ω R belongs to H k (Ω) if and only if v C k 1 (Ω). Even in two dimensions, conforming finite element spaces can be very complicated and there are no satisfactory answer in three dimesnions. Nonconforming approximations are not very popular.

8 Motivation Features of the method Approximation properties Discrete approximations Find u V h such that a(u, v) = l, v for all v V h Cea s lemma u u h V C min v Vh u v V for a constant C independent of h. Requirements for conforming approximations Let k 1 and suppose Ω is bounded. Then a piecewise infinitely differentiable function v : Ω R belongs to H k (Ω) if and only if v C k 1 (Ω). Even in two dimensions, conforming finite element spaces can be very complicated and there are no satisfactory answer in three dimesnions. Nonconforming approximations are not very popular.

9 Motivation Features of the method Approximation properties Stokes equations Find (u, p) H 1 (Ω) n L 2 0 (Ω) such that ν u + p = f in Ω div u = 0 in Ω u = g on Ω Linear elasticity equations Find (σ, u) in H(div, Ω, S) L 2 (Ω, R n ) such that { Aσ = ɛ(u) div σ = f u = g on the boundary and Aσ = 1 2µ σ λ 4µ(µ+λ) tr σ δ, Equilibrium conditions are very difficult to enforce in the finite element method G. Awanou, Spline element method for elasticity, Coming soon

10 Motivation Features of the method Approximation properties Stokes equations Find (u, p) H 1 (Ω) n L 2 0 (Ω) such that ν u + p = f in Ω div u = 0 in Ω u = g on Ω Linear elasticity equations Find (σ, u) in H(div, Ω, S) L 2 (Ω, R n ) such that { Aσ = ɛ(u) div σ = f u = g on the boundary and Aσ = 1 2µ σ λ 4µ(µ+λ) tr σ δ, Equilibrium conditions are very difficult to enforce in the finite element method G. Awanou, Spline element method for elasticity, Coming soon

11 Motivation Features of the method Approximation properties 1 Why multivariate splines for PDEs? Motivation Features of the method Approximation properties 2 Uniform refinement of a tetrahedron Numerical methods for saddle point problems 3 4 Robustness of the method for singular perturbation problems Fourth order singular problem Darcy-Stokes equation 5 Navier-Stokes equations

12 Motivation Features of the method Approximation properties V h = {c R N, Rc = 0}, W h = {c R N, Rc = G} The condition a(u, v) = l, v for all v V h becomes K (c)d = L T d d V h, that is for all d with Rd = 0 K (c, c) + λ T R = L T. [ K T R T R 0 ] [ c λ ] = [ L G ]

13 Motivation Features of the method Approximation properties V h = {c R N, Rc = 0}, W h = {c R N, Rc = G} The condition a(u, v) = l, v for all v V h becomes K (c)d = L T d d V h, that is for all d with Rd = 0 K (c, c) + λ T R = L T. [ K T R T R 0 ] [ c λ ] = [ L G ]

14 Advantages of the method Motivation Features of the method Approximation properties Can be applied to a wide range of PDEs in science and engineering in both two and three dimensions. Constraints and Smoothness are enforced exactly and there is no need to implement basis functions with the required properties. Particularly suitable for higher order PDEs. No inf-sup condition One gets in a single implementation approximations of variable order. The mass and stiffness matrices are assembled easily and this can be done in parallel. Easy implementation of p-adaptive approximation and simplicity of a posteriori error estimate

15 Possible Disadvantages Motivation Features of the method Approximation properties Large size matrices for 3D problems and high order approximations» K T R T R 0» c λ» L = G K T R T R µi c (l+1) λ (l+1)» = L G µλ (l) Computing c (1) from λ (0), one solves (K T + 1 µ RT R)c (l+1) = K T c (l) + 1 µ RT G, l = 1, 2,... c c (l+1) Cµ c c (l)

16 Motivation Features of the method Approximation properties 1 Why multivariate splines for PDEs? Motivation Features of the method Approximation properties 2 Uniform refinement of a tetrahedron Numerical methods for saddle point problems 3 4 Robustness of the method for singular perturbation problems Fourth order singular problem Darcy-Stokes equation 5 Navier-Stokes equations

17 Motivation Features of the method Approximation properties Recall Cea s lemma u u h V C min v Vh u v V Assume V h is a spline space S r d (T h) Bivariate splines For d 3r + 2 and 0 m d and for 0 k m. f Qf p,k Ch m+1 k f p,m+1 The constant C depends on the smallest angle in Trivariate splines For 0 k d, f Wp d+1 (Ω) f Qf p,k Ch d+1 k f p,d+1 The constant C depends on the shape parameter σ K = h K /ρ K

18 Motivation Features of the method Approximation properties Recall Cea s lemma u u h V C min v Vh u v V Assume V h is a spline space S r d (T h) Bivariate splines For d 3r + 2 and 0 m d and for 0 k m. f Qf p,k Ch m+1 k f p,m+1 The constant C depends on the smallest angle in Trivariate splines For 0 k d, f Wp d+1 (Ω) f Qf p,k Ch d+1 k f p,d+1 The constant C depends on the shape parameter σ K = h K /ρ K

19 Uniform refinement of a tetrahedron Numerical methods for saddle point problems 1 Why multivariate splines for PDEs? Motivation Features of the method Approximation properties 2 Uniform refinement of a tetrahedron Numerical methods for saddle point problems 3 4 Robustness of the method for singular perturbation problems Fourth order singular problem Darcy-Stokes equation 5 Navier-Stokes equations

20 Uniform refinement of a tetrahedron Numerical methods for saddle point problems Non degenerate tetrahedron σ T = h T ρ T < Quasi-uniform tetrahedral partition σ T = h T ρ T Care must be taken for uniform refinement σ <, T T h Example of uniform refinement Tetrahedra Sigma Types Tetrahedra Sigma Types

21 Uniform refinement of a tetrahedron Numerical methods for saddle point problems 1 Why multivariate splines for PDEs? Motivation Features of the method Approximation properties 2 Uniform refinement of a tetrahedron Numerical methods for saddle point problems 3 4 Robustness of the method for singular perturbation problems Fourth order singular problem Darcy-Stokes equation 5 Navier-Stokes equations

22 Uniform refinement of a tetrahedron Numerical methods for saddle point problems ( A L T L 0 ) ( c λ λ (l+1) ) = [ F G Assume the system above has a unique solution c. ( ) [ ] [ ] A L T c (l+1) F = L ɛi G ɛλ (l), (2) where λ (0) is a suitable initial guess e.g. λ (0) = 0, Assume that A s = 1 2 (A + AT ) the symmetric part of A is positive definite with respect to L, i.e., x T A s x 0 and x T A s x = 0 with Lx = 0 implies x = 0. Then, the sequence (c (l+1) ) defined by the iterative method converges to the solution c for any ɛ > 0. Furthermore, c c (l+1) Cɛ c c (l) for some constant C independent of ɛ and l. ],

23 Uniform refinement of a tetrahedron Numerical methods for saddle point problems Ac (l+1) + L T λ (l+1) = F and (1) Lc (l+1) ɛλ (l+1) = G ɛλ (l) (2). Multiplying (2) on the left by L T and substituing L T λ (l+1) into (1) and rewriting (2), we get (A + 1 ɛ LT L)c (l+1) = L T λ (l) + F + 1 ɛ LT G (3) λ (l+1) + 1 ɛ Lc(l+1) = λ (l) + 1 ɛ G. A + 1 ɛ LT L is invertible (A + 1 ɛ LT L)x = 0 x = 0.

24 Uniform refinement of a tetrahedron Numerical methods for saddle point problems 0 = x T (A + 1 ɛ LT L)x = x T (A s + 1 ɛ LT L)x = x T A s x + 1 ɛ (Lx)T (Lx) It follows that x T A s x = 0 and (Lx) T (Lx) = 0. so x = 0. c (l+1) converges to c Put u (l+1) = c (l+1) c and p (l+1) = λ (l+1) λ { (A + 1 ɛ LT L)u (l+1) + L T p (l) = 0 p (l+1) = p (l) + 1 ɛ Lu(l+1). p (l) 2 p (l+1) 2 = 2 ɛ (A su (l+1), u (l+1) ) + 1 ɛ 2 Lu(l+1) 2. { p (l) } is seen to be decreasing, bounded below by 0 so cvges Use positive definiteness of A s + 1 ɛ LT L to conclude

25 Uniform refinement of a tetrahedron Numerical methods for saddle point problems We prove that c c (l+1) Cɛ c c (l) { (A + 1 ɛ LT L)u (l+1) + L T p (l) = 0 p (l+1) = p (l) + 1 ɛ Lu(l+1), Au (l+1) + L T p (l+1) = 0 We write u (l+1) = û (l+1) + u (l+1) with û (l+1) Ker(L) and u (l+1) Im(L T ). Note that L : Im(L T ) Im(L) has a bounded inverse, so there exists k 0 > 0 such that from which it follows that u (l+1) 1 k 0 Lu (l+1), u (l+1) 2ɛ k 0 p (l)

26 Uniform refinement of a tetrahedron Numerical methods for saddle point problems To get a bound on û (l+1), we notice that A is invertible on Ker(L) since A + 1 ɛ LT L is invertible. This gives for some α 0 > 0, û (l+1) 1 α 0 sup v 0 Ker(L) (v 0, Aû (l+1) ) v 0 = sup v 0 Ker(L) v T 0 Au(l+1) v 0 A u (l+1). Putting together, we obtain u (l+1) Cɛ p (l), for some constant C > 0 To finish, we need a bound on p (l) in terms of u (l). It can be shown that one can choose λ 0 such that p (l) Im(L) and since L T : Im(L) Im(L T ) has a bounded inverse, p (l) 1 k 0 L T p (l). This completes the proof since L T p (l) = Au (l).

27 Trivariate splines Let d 1 and r 0 S r d (Ω) = {p Cr (Ω), p t P d, t T }. Bijkl d d! (v) = i!j!k!l! bi 1 bj 2 bk 3 bl 4, i + j + k + l = d. The set B d = {B d ijkl (x, y, z), i + j + k + l = d} is a basis for P d. s T = i+j+k+l=d c T ijkl Bd ijkl,

28 Interpolation On the tetrahedron T = v 1, v 2, v 3, v 4 at ξ ijkl = iv 1+jv 2 +kv 3 +lv 4 d On the edge v 1, v 2 at ξ ij = iv 1+jv 2 d c ij Bd ij (v), Bd ij = d! i!j! bi 1 bj 2. p = p = i+j=d i+j+k=d i+j+k+l=d c ijk B d ijk, q = i+j=d c ijkl B d ijkl, q = i+j+k=d c ij0 B d ij0 c ijk0 B d ijk0 Rc = c b

29 Derivatives D i c, i = 1, 2 encode respectively the B-net of s x i. Integration Ω pq = c T Id Smoothness conditions This shows that there s a (l, N) matrix H such that s is in C r (Ω) if and only if Hc = 0.

30 Poisson equation Tetrahedra d=1 d=2 d=3 d= e e e e-02 6*8= e e e e-03 48*8= e e e e-05 Tetrahedra d=5 d= e e-04 6*8= e e-06 u = exp(x + y + z)

31 Biharmonic equation Tetrahedra 6 48 d= e e-02 d= e e-03 d= e e-03 d= e-02 Out of memory u = exp( x 2 y 2 z 2 )

32 Stokes equations u 1 = exp(x + 2y + 3z), u 2 = 2 exp(x + 2y + 3z), u 3 = exp(x + 2y + 3z), p = x(1 x)z(1 z) Table 1 Approximation Errors from Trivariate Spline Spaces on I 1 degrees u 1 u 2 u 3 p Rate d d d d Table 2 Approximation Errors from Trivariate Spline Spaces on I 2 degrees u 1 u 2 u 3 p Rate d d d d

33 Fourth order singular problem Darcy-Stokes equation 1 Why multivariate splines for PDEs? Motivation Features of the method Approximation properties 2 Uniform refinement of a tetrahedron Numerical methods for saddle point problems 3 4 Robustness of the method for singular perturbation problems Fourth order singular problem Darcy-Stokes equation 5 Navier-Stokes equations

34 Biharmonic Poisson Fourth order singular problem Darcy-Stokes equation ɛ 2 2 u u = f in Ω u = 0, u = 0 in Ω n V = W = H0 2 (Ω), ɛ2 u v + u v = Ω V h = {u Sd 1, u = 0 and u/ n = 0 on Ω} {c R N, Hc = 0, Rc = 0, Nc = 0} Ω Ω fv.

35 Fourth order singular problem Darcy-Stokes equation ɛ 2 B + K H T R T N T H R N u 2 = ɛ 2 T T h T c λ 1 λ 2 λ 3 = D 2 u : D 2 v dx + T T h u I h u h u I h T MF Du : Dv dx

36 Fourth order singular problem Darcy-Stokes equation Theorem For d 3r + 2 and 0 m d and for 0 k m. For u H d+1 (Ω), for d 5, f Qf p,k C m+1 k f p,m+1 inf v S 1 d u v 2 ɛ ɛ 2 h 2(d 1) u 2 d+1 + h2d u 2 d+1 = h 2(d 1) (h 2 + ɛ 2 )) u 2 d+1

37 Using cubic polynomials Fourth order singular problem Darcy-Stokes equation ɛ/h Poisson (S) Poisson Biharmonic u(x, y) = (sin(πx) sin(πy)) 2

38 Using polynomials of degree 4 Fourth order singular problem Darcy-Stokes equation ɛ/h Poisson (S) Poisson Biharmonic *

39 Fourth order singular problem Darcy-Stokes equation 1 Why multivariate splines for PDEs? Motivation Features of the method Approximation properties 2 Uniform refinement of a tetrahedron Numerical methods for saddle point problems 3 4 Robustness of the method for singular perturbation problems Fourth order singular problem Darcy-Stokes equation 5 Navier-Stokes equations

40 Fourth order singular problem Darcy-Stokes equation u ɛ 2 u p = f in Ω div u = g in Ω u = 0 on Ω. W = {u H0 1(Ω)2, div u = g} V = {u H0 1(Ω)2, div u = 0} Find u W, ɛ 2 u u + u v = f v, v V. Ω Ω Ω W h = {c R N, Hc = 0, Rc = 0, Dc = G}

41 Fourth order singular problem Darcy-Stokes equation ɛ 2 K + M H T R T D T H R D c λ 1 λ 2 λ 3 = MF 0 0 G v 2 ɛ = v 2 + ɛ 2 v 2 Ṽ = {v L 2 (Ω) 2, div v = 0, v n = 0 on Ω}.

42 Divergence-free vector fields Fourth order singular problem Darcy-Stokes equation V d = {f (H d (Ω)) 2, f = curl φ, φ H d+1 (Ω)} and V d = {s (S 0 d (Ω))2, s = curl S, S S 1 d+1 (Ω)} inf s V d f s 2 h d f d and inf f s 2 h d 1 f d, s V d inf s V d u s ɛ (ɛh d 1 + h d ) u d, d 4. s=0 on Ω

43 Using cubic polynomials Fourth order singular problem Darcy-Stokes equation ɛ/h * Darcy * u = curl` sin(πx) 2 sin(πy) 2, and p = sin(πx)

44 Fourth order singular problem Darcy-Stokes equation Pressure P 3 /P 4 ɛ/h Darcy

45 Using polynomials of degree 4 Fourth order singular problem Darcy-Stokes equation ɛ/h * * Darcy

46 Fourth order singular problem Darcy-Stokes equation Pressure P 4 /P 5 ɛ/h Darcy

47 Navier-Stokes equations 1 Why multivariate splines for PDEs? Motivation Features of the method Approximation properties 2 Uniform refinement of a tetrahedron Numerical methods for saddle point problems 3 4 Robustness of the method for singular perturbation problems Fourth order singular problem Darcy-Stokes equation 5 Navier-Stokes equations

48 Navier-Stokes equations { ν u + 3 j=1 u j u x j + p = f in Ω, div u = 0 in Ω. V 0 = {v H 1 0 (Ω)3, div v = 0} L 2 0 (Ω) = {u L2 (Ω), u = 0} Ω and H 1 2 ( Ω) = {τ(u), u H 1 (Ω)},

49 Navier-Stokes equations Weak formulation : Find u H 1 (Ω) 3 such that ν Ω u v + 3 j=1 Ω u u j v = f v v V 0 x j Ω div u = 0 u = g in Ω on Ω νk c + B(c)c + L T λ = MF Lc = G

50 Lid driven Cavity Flow Navier-Stokes equations FIG.: 3D fluid profile in the x y plane

51 Navier-Stokes equations FIG.: 3D fluid profile in the y z plane

52 Navier-Stokes equations FIG.: 3D fluid profile in the x z plane

53 References Navier-Stokes equations G. Awanou, Energy Methods in 3D Spline Approximations of Navier-Stokes Equations, Ph.d. dissertation, 2003, University of Georgia. G. Awanou, M. J. Lai and P. Wenston, The multivariate spline method for numerical solutions of partial differential equations and scattered data fitting, Wavelets and Splines : Athens 2005, pp G. Awanou and M. J. Lai, Trivariate spline approximations of 3D Navier-Stokes equations, Math. Comp. 74 (2005), pp G. Awanou, Robustness of a spline element method with constraints, Journal of Scientific Computing, 36, 3, (2009) pp