List of Figures

	Figure 1: Function $f = cos3(πx ∕2) + (x + 1)3 ∕8$ (black curve) and its best approximation of degree 2 (red curve). The blue arrows denote the four points where the maximum error is attained.
	Figure 2: Lagrange cardinal polynomials $ℓX 3$ (red curve) and $ℓX 7$ on an uniform grid with $N = 8$ . The black circles denote the nodes of the grid.
	Figure 3: Function $f = cos3(πx ∕2) + (x + 1)3 ∕8$ (black curve) and its interpolant (red curve)on a uniform grid of five nodes. The blue circles show the position of the nodes.
	Figure 4: Function $----1---- f = 1 + 16x2$ (black curve) and its interpolant (red curve) on a uniform grid of five nodes (left panel) and 14 nodes (right panel). The blue circles show the position of the nodes.
	Figure 5: Same as Figure 4 but using a grid based on the zeros of Chebyshev polynomials. The Runge phenomenon is no longer present.
	Figure 6: Function $f = cos3(πx ∕2) + (x + 1 )3 ∕8$ (black curve) and its projection on Chebyshev polynomials (red curve), for $N = 4$ (left panel) and $N = 8$ (right panel).
	Figure 7: Function $f = cos3(πx ∕2) + (x + 1)3∕8$ (black curve) and its interpolant $IN f$ on Chebyshev polynomials (red curve), for $N = 4$ (left panel) and $N = 6$ (right panel). The collocation points are denoted by the blue circles and correspond to Gauss–Lobatto quadrature.
	Figure 8: First Legendre polynomials, from $P 0$ to $P 5$ .
	Figure 9: First Chebyshev polynomials, from $T 0$ to $T 5$ .
	Figure 10: Maximum difference between $f = cos3 (πx∕2) + (x + 1)3∕8$ and its interpolant $I f N$ , as a function of $N$ .
	Figure 11: Step function (black curve) and its interpolant, for various values of $N$ .
	Figure 12: Exact solution (64 ) of Equation (62 ) (blue curve) and the numerical solution (red curve) computed by means of a tau method, for $N = 4$ (left panel) and $N = 8$ (right panel).
	Figure 13: Exact solution (64 ) of Equation (62 ) (blue curve) and the numerical solution (red curve) computed by means of a collocation method, for $N = 4$ (left panel) and $N = 8$ (right panel).
	Figure 14: Exact solution (64 ) of Equation (62 ) (blue curve) and the numerical solution (red curve) computed by means of the Galerkin method, for $N = 4$ (left panel) and $N = 8$ (right panel).
	Figure 15: The difference between the exact solution (64 ) of Equation (62 ) and its interpolant (black curve) and between the exact and numerical solutions for i) the tau method (green curve and circle symbols) ii) the collocation method (blue curve and square symbols) iii) the Galerkin method (red curve and triangle symbols).
	Figure 16: Difference between the exact and numerical solutions of the following test problem. $2 d-u-+ 4u = S dx2$ , with $S (x < 0 ) = 1$ and $S (x > 0) = 0$ . The boundary conditions are $u (x = − 1) = 0$ and $u (x = 1) = 0$ . The black curve and circles denote results from the multidomain tau method, the red curve and squares from the method based on the homogeneous solutions, the blue curve and diamonds from the variational one, and the green curve and triangles from the collocation method.
	Figure 17: Regular deformation of the $[− 1,1] × [− 1,1]$ square.
	Figure 18: Two sets of spherical domains describing a neutron star or black hole binary system. Each set is surrounded by a compactified domain of the type (89 ), which is not displayed
	Figure 19: Definition of spherical coordinates $(r,𝜃,φ)$ of a point $M$ and associated triad $(⃗e ,⃗e ,⃗e ) r 𝜃 φ$ , with respect to the Cartesian ones.
	Figure 20: Regions of absolute stability for the Adams–Bashforth integration schemes of order one to four.
	Figure 21: Regions of absolute stability for the Runge–Kutta integration schemes of order two to five. Note that the size of the region increases with order.
	Figure 22: Eigenvalues of the first derivative-tau operator (124 ) for Chebyshev polynomials. The largest (in modulus) eigenvalue is not displayed; this one is real, negative and goes as $O(N 2)$ .
	Figure 23: Behavior of the error in the solution of the differential equation (135 ), as a function of the parameter $τ$ entering the numerical scheme (136 ).