Chang-Geun Kim and Jungho Park. In this paper, we will consider the following elliptic partial differential equation of order 2.

Transcription

1 Kangweon-Kyungki Math. Jour. 6 (1998), No. 1, pp THE CONDTON NUMBER OF STFFNESS MATRX UNDER p-verson OF THE FEM Chang-Geun Kim and Jungho Park Abstract. n this paper, we briefly study the condition number of stiffness matrix with h-version and analyze it with p-version of the finite element method. 1. Preliminaries n this paper, we will consider the following elliptic partial differential equation of order. (1) u = f in Ω R, Γu = 0 on Γ, where Γu = u or Γu = u n Let Ω R and define Sobolev space H 1 (Ω) = {v L (Ω) v x i L (Ω), i = 1, } Theorem 1.1. Consider the following variational formulation. () F ind u H 1 (Ω) such that a(u, v) = (f, v), for any v H 1 (Ω), where a(u, v) = u v dx and (f, v) = fv dx. Ω Ω Then (1) and () are equivalent,i.e., they have the same solution. Proof. See Johnson [8]. Received December 4, Mathematics Subject Classification: 65N30, 68U0, 68D99. Key words and phrases: Condition number, Finite element method, p-version. The present research has been conducted by the Research Grant of Kwangwoon University in 1997.

2 18 Chang-Geun Kim and Jungho Park Consider a basis of H 1 (Ω). n h-version of the finite element method, we can have the simplest usual basis {ϕ i } M, which is the basis for the finite dimensional space V F = {v H 1 (Ω) : v K P 1 (K), K T h }. On the other hand, in p-version we get the basis {ϕ i } M on the finite dimensional space V F = {v H 1 (Ω) : v K P p [Tp] (Ω), K T p }, where P p [Tp] (Ω) consists of all functions in H 1 (Ω) which are piecewise polynomials of degree at most p and T p is a subdomain of Ω. The notations {ϕ i } M and V F on the following statement are considered h or p-version case. Now, we will derive the approximate solution u A = M a i ϕ i of the problem (1). n the finite dimensional space V F, () becomes the problem: (3) Find u V F such that a(u, v) = (f, v) for any v V F. (4) But, if we choose v = ϕ i for i = 1,..., M, then (3) becomes M a i a(ϕ i, ϕ j ) = (f, ϕ j ), j = 1,..., M which is a linear system of equations in M unknowns a 1,..., a M. n the matrix form, the linear system (4) can be written as (5) Aa = b where A = (a ij ) is the M M matrix with the element a ij = a(ϕ i, ϕ j ), and a = (a 1,..., a M ) and b = (b 1,..., b M ) with b i = (f, ϕ i ).. The Order of the Condition Number n this section we analyze the order of the condition number of the stiffness matrix for the h-version and p-version cases with the model problem given in the section h-version case. Let {ϕ i } M be the usual basis for V h = {v H0(Ω) 1 : v K P 1 (K), K T h },i.e., {ϕ i } M are piecewise polynomials of degree 1. Let a(v, u) = v u dx. And the stiffness matrix A Ω is denoted by (a ij ), where a ij = a(ϕ i, ϕ j ) for i, j = 1,..., M. Then, we have following lemma.

3 The Condition Number of Stiffness Matrix under p-version of the FEM 19 Lemma.1. Let v = M η i ϕ i V h and h = max h K where h K is the longest side of K. Suppose there are positive constants β 1, β such that h K β 1 h and ρ K β. Then, there are constants c and C only h K depending on the constants β 1, β such that and ch η v L Ch η a(v, v) Proof. See Johnson [8]. Ω v dx Ch v L. Theorem.. Let χ(a) be the condition number of A. Then χ(a) = O(h ). Proof. t follows from Lemma.1 and see Johnson [8]. Whatever the finite element spaces are, the above result is not different(see [1]). Moreover, if we take the elliptic problem of order k, then we will have χ(a) = O(h k )(See [8])... p-version case. n this case, we will consider Ω R like h- version case. Let {ψ i } N be the basis of P p [Tp] (Ω), where P p [Tp] (Ω) is the space of all piecewise polynomials on Ω of degree at most p. Lemma.3. Suppose f(x) is a polynomial of degree p. Then, 1 1 Proof. See Bellman [4]. f (x) dx (p + 1)4 1 1 f(x) dx. By Lemma.3, we can prove the following; Lemma.4. Let k be an integer and be an interval in R. Then, for any ψ p P p () where P p () is the set of all polynomials on of degree at most p, ψ p H k () C(k)pk ψ p L (), where C is independent of p and ψ p.

4 0 Chang-Geun Kim and Jungho Park Proof. Using Lemma.3, we get [ ] dψp (p + 1)4 dx ψ p dx (p)4 dx since p 1. Therefore, ψ p H 1 () = [ ψ p + ψ p dx 8p 4 ( ) ] dψp dx Cp 4 ψ p dx. dx Now, the mathematical induction completes the proof. We will state the -dimensional version of Lemma.4 ψ p dx, Lemma.5. Let k be an integer. f S is any triangle, then for each ψ p P p (S), ψ p H k (S) C(k)pk ψ p L (S), where C is independent of p and ψ p. Proof. For each x 1, it follows from Lemma.4 that (6) ψ p (x 1, ) H k () Cp4k ψ p (x 1, ) L (). Similarly for x, (7) ψ p (, x ) H k () Cp4k ψ p (, x ) L (). f we now integrate (6) and (7) with respect to x 1 and x respectively, we get ψ p H k (Q) Cpk ψ p L (Q), where Q =. With an affine mapping, we can infer that Q is a parallelogram. Then, S = m Q i and thus, m ψ p H k (S) ψ p H k (Q i ) Cpk m ψ p L (Q i ) Cpk ψ p L (S). Now we have similar result of the Lemma.1 for the p-versions. Theorem.6. Let v = N a i ψ i, where {ψ i } is the basis. Then, a(v, v) Ω v dx Cp 4 v.

5 The Condition Number of Stiffness Matrix under p-version of the FEM 1 Proof. Take k = 1 on Lemma.5. Then, That is, Thus, ψ p H 1 (S) Cp ψ p L (S), ψ p H 1 (S) = S S where S is a triangle. [ ψp + ψ p ] dx Cp 4 ψ p dx Cp 4 S ψ p dx. S ψ p dx. Let {K i } be subdivision triangles of Ω. Then, with C = sup C i a(v, v) = v dx = (v Ki ) dx Ω K i C i p 4 (v Ki ) dx Cp 4 v dx K i Ω The following theorem gives bounds of the finite element solution. Theorem.7. Let v = N a i ψ i and a = (a 1,..., a N ). Then, v C 1 a, where C 1 is dependent of {ψ i }. Proof. Let M = max { ψ i }. Then we have 1 i N ( N ( ) ( N N v a i ψ i ) M a i = M a i ). ( N ) By the Hölder s inequality, a i K N a i with K is a constant. Thus, N v M K a i. f we denote C 1 = M K, then we have v C 1 N a i = C 1 a.

6 Chang-Geun Kim and Jungho Park Theorem.8. Let v = N a i ψ i. Then, there exists a constant C such that C a v. Proof. Using that a + b + a b = ( a + b ), we have N 1 N 1 N 1 a i ψ i + a N ψ N + a i ψ i a N ψ N = a i ψ i + a N ψ N, i N 1 a i ψ i + a N 1 ψ N 1 + a i ψ i a N 1 ψ N 1 i N 1 = a i ψ i + a N 1 ψ N 1,. i N 1 a i ψ i + a 1 ψ 1 + a i ψ i a 1 ψ 1 i 1 i 1 = a i ψ i + a 1 ψ 1. i 1 f we sum right and left hands of the above equalities respectively, we get N N N a i ψ i + a i ψ i a k ψ k = So, we see that i k N a k ψ k N N a i ψ i + N a i ψ i + a k ψ k. i k N a i ψ i a k ψ k. i k

7 The Condition Number of Stiffness Matrix under p-version of the FEM 3 We denote A by N N a i ψ i for efficiency. Thus, N a k ψ k A + A + N a i ψ i a k ψ k i k ( N a i ψ i + a k ψ k i k ) ( N ( N = A + N a i ψ i ) = A + N a i ψ i ( N A + NM a i ) A + NM K ) N a i. Here M = max ψ i and the last inequality follows by the Hölder s 1 i N inequality. Therefore, we have N a k ψ k N N N a i ψ i + NM K a i. So, (8) N a k ψ k N N a i ψ i + NM K N a i. Using (8) with L = min 1 i N ( ψ i ), we have L N a i N a k ψ k N N a i ψ i + NM K N a i. Let C = (L NM K)/N. Then we get the following inequality which completes the proof. N C a i N a i ψ i = v.

8 4 Chang-Geun Kim and Jungho Park Now, we will prove the main theorem. This theorem is parallel to Theorem., that is, the following theorem is a version of Theorem. for p-version case. Theorem.9. Let {ψ i } be the basis of P p [Tp] (Ω). Let A be the stiffness matrix, i.e., A = (a ij ), where a ij = a(ψ i, ψ j ) = ψ Ω i ψ j dx. And let χ(a) be the condition number of A. Then, χ(a) = O(p 4 ). Proof. Let v = N a i ψ i. Then, a(v, v) = a Aa. Using Theorem.6 and Theorem.7, we have (9) a Aa a = a(v, v) a On the other hand, by Theorem.8 (10) a Aa a = But we know that λ max = max a 0 a(v, v) a Cp 4 v a Cp4 C 1 a a = CC 1 p 4, α v a αc a a = αc. a Aa a Aa and λ a min = min a 0 a. Therefore, we get λ max CC 1 p 4 and λ min αc by (9) and (10). Hence χ(a) = λ max λ min CC 1 αc p 4, where α, C 1, C and C are constants. This completes the proof. n the case ( u + u = f), the result and the proof are quite similar to those of Theorem.9. Roughly speaking, elliptic problem of order 4 will have the order p 8 and p-version has order twice h-version in the condition number. 3. Numerical Experiment We consider the following problem: { u (x) = x/ 1 x for x Ω = ( 1, 1), u( 1) = π 4, u(1) = π 4.

9 The Condition Number of Stiffness Matrix under p-version of the FEM 5 Table 1. Eigenvalues and Condition Numbers p λ max λ min Condition Number n this case, the bilinear product is a(u, v) = 1 1 u (x)v (x) dx. By the variational formulation, we seek a solution u H 1 (Ω) which satisfies a(u, v) = 1 1 u (x)v (x) dx = 1 1 x v(x) dx 1 x for all v H 1 (Ω). To get the approximate solution of the finite element method, we ought to take some basis functions. We choose as basis functions ψ 0 (x) = 1 x, ψ 1 (x) = x and ψ k+1 (x) = x 1 P k (t) dt for k 1, where P k (t) is the monic Legendre polynomial of degree k. n this paper we analyze the condition number of stiffness matrix and so we will find it in place of the approximate solution. The procedure is programmed using Turbo C, including the Gaussian quadrature (See [5]) with n = 5 for numerical integration and the Jacobi s method for finding eigenvalues of the stiffness matrix. The result is shown in Table 1 with maximum degree of basis functions p = 1,, 3, 4, 5. The maximum, minimum eigenvalues and the condition numbers of stiffness matrices are given in Table 1. The result has coincidence with Theorem.9. As the degree p grows, the condition number is larger rapidly. Since we use the Gaussian quadrature with n = 5, the approximation of integration is exact if the integrand has degree at most 9 except roundoff error. Thus the approximate values of integrations are quite exact in this case because the integrands are polynomials with degree from 0 to 8. Also to deal with a non-singular stiffness matrix we take p p matrix because of ψ 0 = 1 while one gets generally (p + 1) (p + 1) matrix as the stiffness matrix.

10 6 Chang-Geun Kim and Jungho Park References [1] O. Axelssen and V. A. Barker. Finite Element Solution of Boundary Value Problems. Academic Press, nc., Florida,1984. []. Babuška and B. Szabo. Finite Element Analysis. John Wiley & Sons, nc., New York, [3]. Babuška, B. A. Szabo, and. N. Katz. The p-version of the finite element method. SAM J. Numer. Anal., 18: , [4] R. Bellman. A note on an inequality of e. schmidt. Amer. Math. Soc., 50: , [5] R. L. Burden and J. D. Faires. Numerical Analysis. PWS-KENT,Boston, [6] P. G. Ciarlet. The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, [7] M. R. Dorr. The approximation theory for the p-version of the finite element method. SAM J. Numer. Anal., 1: , [8] C. Johnson. Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, New York,1987. [9] M. Krizek and P. Neittaanmäki. Finite Element Approximation of Variational Problems and Applications. Longman Scientific & Technical, England,1990. Chang-Geun Kim Department of Mathematics Research nstitute of Basic Science, Kwangwoon University Seoul , Korea ckim@@daisy.kwangwoon.ac.kr Jungho Park Department of Mathematics, Yonsei University, Seoul , Korea