Variable-step preconditioned conjugate gradient method for partial ...

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[31] K. Stuben, An Introduction to Algebraic Multigrid, In the book U. Trottenberg, C. Oosterlee and A. Schuller, Multigrid, Academic Press, 2001. [32] U. Trottenberg, C. Oosterlee and A. Schuller, Multigrid, Academic Press, 2001. Appendix A To investigate the strange behavoiour of the convergance rate in Tables 1 and 2 we have to investigate the formula for reduction factor q(εin, λkj , λkj+1)). Since for a fixed value of εin the number of iterations depends only on the eigenvalues ratio, which is for discretization matrices depend on the characteristic meshsize h. Note that it is a constant for the continue case. For simplisity of presentation we consider the two-dimensional case. Formulas both for continous {λl,m, v (l,m) } and for discret eigenpairs {λh l,m , v(l,m) } can be easily derived, and they are and λl,m = π 2 � a11l 2 + a22m 2� , v (l,m) = cos (lπx) cos (mπy) , h l, m = 1, 2, . . . , λ h l,m = 4 h2 � a11 sin 2 � πh 2 l � + a22 sin 2 � πh 2 m �� , [v (l,m) h ] i,j = v (l,m) (ih, jh), l, m = s1, . . . , N = h −1 . Hence, in the case a11 = a22 = 1 the first and second eigenvalues are Denoting αh = πh 2 λ h 1 = λ h 1,1 = 4 h 2 λ h 2 = λ h 1,2 = 4 h 2 � sin 2 � sin 2 � πh 2 � πh 2 � + sin 2 � + sin 2 � πh 2 � πh 2 2 �� , �� . and using well-known trigonometric formulae one can easyly derived that λh 1 λh 2 = 2 1 + 4 cos 2 αh Now consider two meshes with different meshsize h and H > h and compare their eigenvalues ratio. Since the function f(x) = cos x is a monotonical decreasing function on the interval [0, π/2] we obtain Thus, we proof that λh 1 λh 2 = 2 1 + 4 cos 2 αh q(εin, λ h 1, λ h 2) = εin + (1 − εin) λh 1 λ h 2 < 2 1 + 4 cos 2 αH . = λH 1 λ H 2 < εin + (1 − εin) λH 1 λ H 2 . = q(εin, λ H 1 , λ H 2 ), and hence, the number of iterations is decreased when h goes to zero, and the limit number of iteration is bounded by the ratio (λ1/λ2) for continue eigenvalues. The similar behaviour can be expected in the general case for elliptic BVPs. 18 (19)
Appendix B Without loss of generallyty, we can assume that a11 = 1 and a22 = ɛ ≤ a11. As it is readily seen from (19) the first and second eigenvalues are λ h 1 = λ h 1,1 = 4 h2 � sin 2 � � πh + ɛ sin 2 2 � �� πh , 2 λ h 2 = λ h 1,2 = 4 h2 � sin 2 � � πh + ɛ sin 2 2 � πh 2 2 �� , and hence, we obtain q(εin, λ h 1, λ h 2) = εin + (1 − εin) λh 1 λ h 2 On the other side for a22 = δ < ɛ we have q(εin, ˆ λ h 1, ˆ λ h 2) = εin + (1 − εin) ˆ λ h 1 ˆλ h 2 = = 1 + ɛ 1 + 4ɛ cos 2 αh 1 + δ 1 + 4δ cos 2 αh + (4 cos2 αh − 1)εinɛ 1 + 4ɛ cos2 . αh + (4 cos2 αh − 1)εinδ 1 + 4δ cos2 . αh Since αh is close to zero, then in what follow we can use cos 2 αh ≈ 1. Thus, we obtain q(εin, λ h 1, λ h 2) − q(εin, ˆ λ h 1, ˆ λ h 2) = 1 + ɛ 1 + δ · 1 + 4ɛ 1 + 4δ · 3 · (1 + εin) · (ɛ − δ). which tends to zero as ɛ when ɛ → 0 and δ → 0, and hence, the number of iterations does not considerably changed with respect to the anisotropy ratio. 19
Page 1 and 2: Variable-step preconditioned conjug
Page 3 and 4: which is too complicated for unders
Page 5 and 6: Theorem 1 shows that a certain sequ
Page 7 and 8: Algorithm LOBPCG-VS Input: m starti
Page 9 and 10: 6 Numerical results All numerical r
Page 11 and 12: Since all results in Table 1 and 2
Page 13 and 14: Table 7. Number of iterations for L
Page 15 and 16: Table 9. Number of iterations for L
Page 17: [11] A.V. Knyazev, New estimates fo

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