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180 Polynomial Approximation of Differential Equations in (7.4.9). This is obtained starting from the set of grid-points η (n) j , 0 ≤ j ≤ n (finegrid). Another set of points in [−1,1] (coarse-grid) is given by the nodes η (k) j , 0 ≤ j ≤ k, where 2 ≤ k ≤ n. A polynomial p ∈ Pn, determined by its values on the fine-grid, is evaluated at the coarse-grid by using formula (3.2.7). On the other hand, we can go from the coarse-grid to the fine-grid by extrapolation, i.e. k (8.7.1) p(η (n) i ) = j=0 p(η (k) j ) ˜ l (k) j (η (n) i ), 0 ≤ i ≤ n. In short, the philosophy of the multigrid method is the following. Since the eigenvalues of D (n) are not available (the cost of their computation is very high), an alternative to varying the parameter θ is to accelerate the convergence by using different grids. A good treatment of low modes is obtained by applying the Richardson method to the matrix D (k) (k small) defined on the coarse-grid. Then, we can switch to the fine-grid by (8.7.1), and work with the matrix D (n) to damp the high modes with a relatively small θ. In general, the algorithm applies to different sets of grid-points, selected in turn, according to some prescribed rule (the most classical one is the V- cycle). There are many ways to advance the method. These depend on the problem and the competence of the user. A comparative analysis is given in heinrichs(1988). Other results are provided in muñoz(1990). Early applications within the framework of spectral methods, for more specialized problems, are examined, for instance, in zang, wong and hussaini(1982), streett, zang and hussaini(1985), brandt, fulton and taylor(1985), zang and hussaini(1986). For a general overview of the method, we refer to hackbusch(1985). Using the multigrid method for the preconditioned systems in place of a plain preconditioned iterative scheme, does not improve very much the rate of convergence for problems in one dimension. More interesting are the applications in two or three dimensions. Performances also depend on the velocity and the cost to transfer the data from one grid to the next. Particular efficiency is obtained in the Chebyshev case. When n is even and k = n/2, recalling relation (3.8.16), all the points of the coarse-grid are also points of the fine-grid. Furthermore, the inverse formula (8.7.1) is implemented via FFT.
9 ORDINARY DIFFERENTIAL EQUATIONS The large amount of material accumulated in the previous chapters is now directed towards the study of the convergence of approximate solutions to differential equations. Beginning with the simplest linear equations, we then extend our analysis to other, more complex, problems. Throughout this chapter, however, we remain focused on different techniques, rather than on solving complicated equations. 9.1 General considerations Assume that I ⊂ R is a bounded open interval. Without loss of generality we set I :=] − 1,1[. In fact, any other interval is mapped into ] − 1,1[ by a linear change of variables and a translation. Note that the space of polynomials Pn, n ∈ N, is invariant under these transformations. Let f : Ī → R be a given continuous function. A very simple differential equation is (9.1.1) ⎧ ⎨ U ′ = f in ] − 1,1], ⎩ U(−1) = σ,
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PREFACE This book is devoted to the
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CONTENTS 1. Special families of pol
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Contents ix 7. Derivative Matrices
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1 SPECIAL FAMILIES OF POLYNOMIALS A
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Special Families of Polynomials 3 n
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Special Families of Polynomials 5 B
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Special Families of Polynomials 7 1
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Special Families of Polynomials 9 (
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Special Families of Polynomials 11
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2 ORTHOGONALITY Inner products and
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Orthogonality 23 The function w is
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Orthogonality 25 As we promised, we
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Orthogonality 27 Let us now define
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Orthogonality 29 Therefore, one obt
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Orthogonality 31 From (2.3.8), we g
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Orthogonality 33 only mention the r
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3 NUMERICAL INTEGRATION As we shall
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Numerical Integration 37 In the cas
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Numerical Integration 39 (3.1.9) z
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Numerical Integration 41 We give th
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Numerical Integration 43 where 1
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Numerical Integration 45 (3.2.10)
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Numerical Integration 47 This means
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Numerical Integration 49 (3.4.2) w
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Numerical Integration 51 3.5 Gauss-
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Numerical Integration 53 The proof
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Numerical Integration 55 3.7 Clensh
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Numerical Integration 57 the first
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Numerical Integration 59 (3.8.14) p
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Numerical Integration 61 (3.9.5) p
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Numerical Integration 63 (3.10.5)
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66 Polynomial Approximation of Diff
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100 Polynomial Approximation of Dif
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REFERENCES adams r.(1975), Sobolev
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References 295 canuto c., hussaini
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References 297 friedman a.(1959), F
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References 299 jackson d.(1930), Th
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References 301 pavoni d.(1988), Sin
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References 303 vainikko g.m.(1964),
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