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19 Mar 2020: 23 24 25 26 270. 0.5. 1. 1.5. cross-sectional view side-view.

23 Nov 2020: Time, location: Online. This is a Part II course in numerical analysis given in Michaelmas term 2020 that contains 24 lectures. ... Lecture 16:Lecture 17:Lecture 18:Week 7:. Lecture 19:Lecture 20:Lecture 21:Week 8:. Lecture 22:Lecture 23:Lecture

6 Oct 2020: 24. The QR algorithm. Convergence of the first column and the bottom row.

29 Oct 2020: 19 on the Example Sheets). Method 3.24 (The spectral method for evolutionary PDEs) We consider the problemu(x,t). ... û′n(t) = π2n2ûn(t). n ΓN (3.24). Its exact solution is ûn(t) = eπ2n2t ĝn for n 6= 0 and we set û0(t) = 0 due to

9 Nov 2020: criterion. So, you need also to prove that ρ(D1(AD)) > 1.]. 24. ... Further, show that if one Jacobi iteration isperformed, then u3,3 = 23/24 occurs, which is the estimate of u(1/2,1/2) = 9/8.

15 Oct 2020: 24 / 26. Stability: Backward Euler and the diffusion equation. 2) For the backward Euler method.

26 Oct 2020: 24. the even and odd portions of y, respectively. Note that y(E),y(O) Πm.

16 Nov 2020: x n1} using a recurrencerelation. 24 / 26. Conjugate gradient - Warm up.

16 Nov 2020: This gives expression for αk in (4.7). 37. Method 4.24 (The steepest descent method) This method takes d(k) =F1(x(k)) = b Ax(k)for every k, the

15 Oct 2020: This, however,does not mean that its numerical modelling is easy. 13 / 24. ... 23 / 24. Stability using Fourier analysis. The Fourier analysis (for Cauchy problem) provides.

23 Nov 2020: Numerical Analysis - Part II. Anders C. Hansen. Lecture 24. 1 / 25. ... 24 / 25. Single shifts. A good approximation sk = (Ak )nn to the eigenvalue λs generateseven better approximation of sk1 = (Ak1)nn to λs , andconvergence is accelerating at a

26 Oct 2020: etBetC = et(BC) 12t2(BC CB) O(t3). (2.24). In particular, etBetC = et(BC) for all t 0 if and only if B and C commute.

24 Nov 2020: δmn. = α(n)(t) , (1.24). reproducing (1.23) apart from renaming the indices. ... 1/23/21/2. . . . . . . . . (1.30). 24.

5 Oct 2020: 24. The solution is. u = 12µ. dp. dx(h2 y2). The volume flux (per unit length in z-direction) is.

9 Oct 2020: 24 / 27. Crank–Nicolson for advection equation. Let. un1m unm = 14µ(un1m1u.

26 Oct 2020: N/2. (12). 24 / 27. Computation of Fourier coefficients (DFT).

9 Nov 2020: Multigrid in practice. Figure : Wikipedia’s interpretation of multigrid. 24 / 30.

29 Oct 2020: 19 on the ExampleSheets). 24 / 27. The spectral method for evolutionary PDEs.

28 Oct 2020: 24 / 29. Analyticity and periodicity – Relaxing periodicity. Relaxing periodicity: Disappointingly, periodicity is necessary forspectral convergence.

21 Oct 2020: 24 / 28. Splitting of inhomogeneous systems. Recall our goal, namely fast methods for the two-dimensionaldiffusion equation.

12 Nov 2020: 24 / 29. The Krylov subspaces. Proof. Inequality (7) is true not just for positive definite A > 0, butfor any A with n linearly independent eigenvectors (ui ). Indeed, inthat case one can

2 Nov 2020: 23 / 29. Iterative methods for linear algebraic systems. 24 / 29.

16 Nov 2020: k=0. αkd(k). 24 / 27. Rate of convergence of CGM. Proof of Theorem 7.

19 Oct 2020: 24 / 34. Special structure of 5-point equations. Observation 5 (Special structure of 5-point equations).

19 Nov 2020: In general, the cost of thisprocedure is O(n3). 24 / 27.

6 Dec 2020: The SCI provides a classificationhierarchy [8,9,24–27,42] of spectral problems according to their computational difficulty.

5 Nov 2020: 24 / 36. Poisson’s equation on a square. Note that when p or q is equal to 1 or m, then the values u0,q, up,0or up,m1, um1,q

1 May 2020: 24, Section 5.16]). Let X ′ X be a closed subspace. ... j=1. 〈f,yj〉Y2σ2j. < (2.11). is met. 24 CHAPTER 2. GENERALISED SOLUTIONS.