Numerical Solutions to Partial Differential Equations

Transcription

1 Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University

2 A Framework for Interpolation Error Estimation of Affine Equivalent FEs 1 The polynomial quotient spaces of a Sobolev space and their equivalent quotient norms ((2) in the example); 2 The relations between the semi-norms of Sobolev spaces defined on affine-equivalent open sets ((3), (4) in the exmp.); 3 The abstract error estimates for the polynomial invariant operators ((1) in the example); 4 To estimate the constants appeared in the relations of the Sobolev semi-norms by means of the geometric parameters of the corresponding affine-equivalent open sets. 2 / 23

3 The Semi-norm v k+1,p,ω Is an Equivalent Norm of W k+1,p (Ω)/P k (Ω) 1 The quotient space W k+1,p (Ω)/P k (Ω) is a Banach space. 2 The quotient norm of a function v is defined by v W k+1,p (Ω)/P k (Ω) v k+1,p,ω := inf w P k (Ω) v+w k+1,p,ω. 3 v W k+1,p (Ω)/P k (Ω) v k+1,p,ω = v k+1,p,ω is a semi-norm, and obviously v k+1,p,ω v k+1,p,ω. Theorem There exists a constant C(Ω) such that v k+1,p,ω C(Ω) v k+1,p,ω, v W k+1,p (Ω)/P k (Ω).

4 Relations of Sobolev Semi-norms on Affine Equivalent Open Sets Extension of the 1-D Result to the General Case Relations of Semi-norms on Open Sets Related by F (ˆx) = hˆx + b R n 1 Let F : ˆx R n F (ˆx) = hˆx + b R n, and Ω = F (ˆΩ) 2 Then, α v(x) = h α αˆv(ˆx), and dx = det(b) dˆx = h n dˆx. 3 Therefore, by a change of integral variable, we have v m,p,ω = h m det(b) 1/p ˆv m,p,ˆω = h m+n/p ˆv m,p,ˆω. 4 diam(ω)/diam(ˆω) = h v m,p,ω / ˆv m,p,ˆω h m+n/p. 4 / 23

5 Relations of Sobolev Semi-norms on Affine Equivalent Open Sets Extension of the 1-D Result to the General Case Affine Equivalent Open Sets Related by F (ˆx) = Bˆx + b R n Let Ω = F (ˆΩ) be affine equivalent open set in R n with F : ˆx R n F (ˆx) = Bˆx + b R n, For v W m,p (Ω) and ˆv(ˆx) = v(f (ˆx)), the Sobolev semi-norms v m,p,ω and ˆv m,p,ˆω have a similar relation for general B, i.e. where h = diam(ω)/diam(ˆω). v m,p,ω / ˆv m,p,ˆω h m+n/p, 5 / 23

6 Relations of Sobolev Semi-norms on Affine Equivalent Open Sets Extension of the 1-D Result to the General Case Relations of Semi-norms on Open Sets Related by F (ˆx) = Bˆx + b Theorem Let Ω and ˆΩ be two affine equivalent open sets in R n. Let v W m,p (Ω) for some p [1, ] and nonnegative integer m. Then, ˆv = v F W m,p (ˆΩ), and there exists a constant C = C(m, n) such that ˆv m,p,ˆω C B m det(b) 1/p v m,p,ω, where B is the matrix in the affine mapping F, represents the operator norms induced from the Euclidian norm of R n. Similarly, we also have v m,p,ω C B 1 m det(b) 1/p ˆv m,p,ˆω. 6 / 23

7 Relations of Sobolev Semi-norms on Affine Equivalent Open Sets Extension of the 1-D Result to the General Case Proof of ˆv m,p,ˆω C(n, m) B m det(b) 1/p v m,p,ω 1 Let ξ i = (ξ i1,..., ξ in ) T R n, i = 1,, m, be unit vectors, D = ( 1,..., n ), D mˆv(ˆx)(ξ 1,..., ξ m ) = ( m i=1 D ξ i) ˆv(ˆx). 2 Assume v C m ( Ω), therefore, ˆv C m (ˆΩ) also. We have αˆv(ˆx) D mˆv(ˆx) := sup D mˆv(ˆx)(ξ 1,..., ξ m ), α = m. ξ i =1 1 i m 3 Let C 1 (m, n) be the cardinal number of α, then ˆv m,p,ˆω = ( ˆΩ α =m αˆv(ˆx) 1/p ( dˆx) p C1 (m, n) D mˆv(ˆx) 1/p dˆx) p. ˆΩ 7 / 23

8 Proof of ˆv m,p,ˆω C(n, m) B m det(b) 1/p v m,p,ω 4 On the other hand, by the chain rule of differentiations for composition of functions, F (ˆx) (D ξ)ˆv(ˆx) = D(v F (ˆx)) ξ = Dv(x) ξ = (D Bξ)v(x). ˆx 5 Therefore, ( m i=1 D ξ i) ˆv(ˆx) = ( m i=1 D Bξ i) v(x), i.e. D mˆv(ˆx)(ξ 1,..., ξ m ) = D m v(x)(bξ 1,..., Bξ m ). 6 Consequently, D mˆv(ˆx) B m D m v(x). 7 Thus, by a change of integral variable, we obtain D mˆv(ˆx) p dˆx B mp ( det B 1 ) D m v(x) p dx. ˆΩ Ω

9 Relations of Sobolev Semi-norms on Affine Equivalent Open Sets Extension of the 1-D Result to the General Case Proof of ˆv m,p,ˆω C(n, m) B m det(b) 1/p v m,p,ω 8 For any given η i R n with η i = 1, 1 i m, we have [ m n n m D m v(x)(η 1,..., η m ) = η ij j v(x) = η iji ji ]v(x). i=1 j=1 j 1,,j m=1 i=1 9 Since, η ij 1, 1 i m, 1 j n, we are lead to D m v(x) n m max α =m α v(x) n m( α =m α v(x) p) 1/p. 9 / 23

10 Relations of Sobolev Semi-norms on Affine Equivalent Open Sets Extension of the 1-D Result to the General Case Proof of ˆv m,p,ˆω C(n, m) B m det(b) 1/p v m,p,ω 10 By 3, 7 and 9, the inequality hold for v C m ( Ω). 11 For 1 p <, C m ( Ω) is dense in W m,p (Ω), so the inequality also holds for all v W m,p (Ω). 12 If p =, since the inequality holds uniformly for 1 q <, and for the bounded domain Ω w 0,,Ω = lim q w 0,q,Ω, w L (Ω), therefore, the inequality holds also for v W m, (Ω). 10 / 23

11 Relations of Sobolev Semi-norms on Affine Equivalent Open Sets Estimate B and det(b) by Geometric Parameters Bound B and B 1 by the Interior and Exterior Diameters 1 Denote the exterior and interior of a region Ω as { h Ω := diam (Ω), ρ Ω := sup {diam (S) : S Ω is a n-dimensional ball}. Theorem Let Ω and ˆΩ be two affine-equivalent open sets in R n, let F (ˆx) = Bˆx + b be the invertible affine mapping, and Ω = F (ˆΩ). Then, B hˆρ, and B 1 ĥ ρ, where h = h Ω, ĥ = hˆω, ρ = ρ Ω, ˆρ = ρˆω. 11 / 23

12 Relations of Sobolev Semi-norms on Affine Equivalent Open Sets Estimate B and det(b) by Geometric Parameters Proof of B hˆρ and the Geometric Meaning of det(b) 1 By the definition of B, we have B = sup Bξ. 1ˆρ ξ =ˆρ 2 Let the vectors ˆx, ŷ ˆΩ be such that ŷ ˆx = ˆρ, then, we have x = F (ˆx) Ω, y = F (ŷ) Ω. 3 Therefore, B(ŷ ˆx) = F (ŷ) F (ˆx) h B hˆρ. The determinant det(b) also has an obvious geometric meaning: det(b) = meas(ω) meas( ˆΩ) and det(b 1 ) = meas(ˆω) meas(ω). 12 / 23

13 A Result on Affine Equivalent Family of Polynomial Invariant Operators Theorem Let nonnegative integers k, m and p, q [1, ] be such that W k+1,p (ˆΩ) W m,q (ˆΩ). Let the bounded linear operator ˆΠ L(W k+1,p (ˆΩ); W m,q (ˆΩ)) be P k (ˆΩ) invariant, meaning ˆΠŵ = ŵ, ŵ P k (ˆΩ). Let Ω = F (ˆΩ) be an arbitrary open set which is affine equivalent to ˆΩ, where F (ˆx) = Bˆx + b. Let the linear operator Π Ω L(W k+1,p (Ω); W m,q (Ω)) be defined by ) Π Ω v = (ˆΠ (v F ) F 1. Then, there exists a constant C = C(ˆΠ, ˆΩ, n, m, k) independent of Ω, such that, for all v W k+1,p (Ω), v Π Ω v m,q,ω C (meas(ω)) (1/q 1/p) hk+1 Ω ρ m Ω v k+1,p,ω.

14 Proof of Error Estimate for Affine-Family of Polynomial Invariant Operators 1 By the polynomial P k (ˆΩ) invariant of ˆΠ, we have ˆv ˆΠˆv = (I ˆΠ)(ˆv +ŵ), ˆv W k+1,p (ˆΩ), ŵ P k (ˆΩ). 2 By W k+1,p (ˆΩ) W m,q (ˆΩ), ˆΠ L(W k+1,p (ˆΩ); W m,q (ˆΩ)), and the semi-norm k+1,p,ˆω is an equivalent quotient norm: ˆv ˆΠˆv m,q,ˆω I ˆΠ inf ˆv+ŵ C(ˆΠ, k+1,p,ˆω ˆΩ) ˆv. k+1,p,ˆω ŵ P k (ˆΩ) 3 Since v Π Ω v = (ˆv ˆΠˆv) F 1, by Theorem 7.3, we have v Πv m,q,ω C(m, n) B 1 m det(b) 1/q ˆv ˆΠˆv m,q,ˆω, ˆv k+1,p,ˆω C(k + 1, n) B k+1 det(b) 1/p v k+1,p,ω. 4 The estimate follows as a consequence of the above three inequalities and Theorem 7.4.

15 Error Estimates of Affine Family Finite Element Interpolations Theorem Let ( ˆK, ˆP, Σ) be a finite element, let s be the highest order of the partial derivatives appeared in the set of the degrees of freedom Σ. Let nonnegative integers k and m, and p, q [1, ] be such that W k+1,p ( ˆK) C s ( ˆK), W k+1,p ( ˆK) W m,q ( ˆK), P k ( ˆK) ˆP W m,q ( ˆK). Then, there exists a constant C( ˆK, ˆP, Σ) such that, for all finite elements (K, P, Σ) which are affine equivalent to ( ˆK, ˆP, Σ), and for all v W k+1,p (K), v Π K v m,q,k C( ˆK, ˆP, Σ) (meas(k)) (1/q 1/p) hk+1 K ρ m K v k+1,p,k.

16 Error Estimates of Polynomial Invariant Operators Error Estimation for Finite Element Interpolation Operators Proof of Error Estimates for Affine-Family of Finite Element Interpolations 1 By the error estimates for affine-family of polynomial invariant operators (see Theorem 7.5), we only need to verify that the corresponding finite element interpolation operators ˆΠ = ˆΠ ˆK L(Wk+1,p ( ˆK); W m,q ( ˆK)). 2 Let {ŵ i } N i=1 be a set of basis of ˆP, and { ˆϕ i } N i=1 Σ be the corresponding dual basis. 3 { ˆϕ i } N i=1 are also bounded linear functionals on Cs ( ˆK), since s is the highest order of partial derivatives in Σ. 16 / 23

17 Error Estimates of Polynomial Invariant Operators Error Estimation for Finite Element Interpolation Operators Proof of Error Estimates for Affine-Family of Finite Element Interpolations 4 By ˆΠˆv = N i=1 ˆϕ i(ˆv)ŵ i ˆP W m,q ( ˆK), and W k+1,p ( ˆK) C s ( ˆK), we have, for all ˆv W k+1,p ( ˆK), N ˆΠˆv m,q, ˆK ˆϕ i (ˆv) ŵ i m,q, ˆK i=1 C ( N ) ŵ i m,q, ˆK ˆv s,, ˆK C 1 ˆv k+1,p, ˆK. i=1 17 / 23

18 Error Estimates of Polynomial Invariant Operators Error Estimation for Finite Element Interpolation Operators Relations Between Geometric Parameters of a Finite Element K Denote σ n = meas{x R n : x 1}, we have σ n ρ n K meas (K) hn K. In the finite element error analysis, the convergence speed can be characterized by the powers of the element K s diameter h K, if the finite elements satisfy certain geometric conditions, for example, the regularity condition given below. 18 / 23

19 Error Estimates of Polynomial Invariant Operators Error Estimation for Finite Element Interpolation Operators Regular Family of Finite Element Triangulations Definition {T h (Ω)} h>0 is said to be a regular family of finite element triangulations of Ω, if (i) there exists a constant σ independent of h such that h K σ ρ K, K h>0 T h (Ω); (ii) 0 is the accumulation point of the parameter h. 19 / 23

20 Error Estimates of Polynomial Invariant Operators Error Estimation for Finite Element Interpolation Operators Regular Affine Equivalent Family of Finite Element Triangulations Let {T h (Ω)} h>0 be a regular family of finite element triangulations of Ω. Let {(K, P K, Σ K )} K h>0 T be a family of finite elements, h(ω) each of which is affine equivalent to the reference finite element ( ˆK, ˆP, ˆΣ). Then, the finite elements {(K, P K, Σ K )} K h>0 T h(ω) regular affine equivalent family of finite elements. are called a 20 / 23

21 Error Estimates of Polynomial Invariant Operators Error Estimation for Finite Element Interpolation Operators Error Estimates of Regular Affine Family of Finite Element Interpolations Theorem Let {T h (Ω)} h>0 be a regular family of finite element triangulations of Ω, let {(K, P K, Σ K )} K h>0 T be a regular affine equivalent h(ω) family of finite elements with respect to the reference finite element ( ˆK, ˆP, ˆΣ), which satisfies the conditions (7.2.11) (7.2.13) of Theorem 7.6 (Them. on interpolation error of affine equivalent FEs). Then, there exists a constant C = C( ˆK, ˆP, ˆΣ) such that v Π K v m,q,k C (meas (K)) (1/q 1/p) σ m h k+1 m K v k+1,p,k, v W k+1,p (K), K h>0 T h (Ω), where σ is the constant in the definition of regularity. 21 / 23

22 Error Estimates of Polynomial Invariant Operators Error Estimation for Finite Element Interpolation Operators Proof of Error Estimates for Regular Affine Family of Finite Elements 1 By the error estimates for affine equivalent family of finite element interpolation operators (Theorem 7.6), we have that, for all v W k+1,p (K), v Π K v m,q,k C( ˆK, ˆP, Σ) (meas(k)) (1/q 1/p) hk+1 K ρ m K 2 By the regularity of the triangulation, we have ρ 1 K σh 1 K, K h>0 T h (Ω). v k+1,p,k. 3 Thus, the conclusion of the theorem follows. 22 / 23

23 SK 7µ5, 6. Thank You!