2006 | OriginalPaper | Chapter
Compactly Supported Fundamental Functions for Spline-Based Differential Quadrature
Authors : Domingo Barrera Rosillo, Francisco Ibáñez Pérez
Published in: III European Conference on Computational Mechanics
Publisher: Springer Netherlands
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The Differential Quadrature Method (DQM) is a numerical discretization technique for the approximation of derivatives by means of weighted sums of function values. It was proposed by Bellman and coworkers in the early 1970’s, and it has been extensively employed to approximate spatial partial derivatives (cf. [
1
], [
4
] for instance). The classical DQM is polynomial-based, and it is well known that the number of grid points involved is usually restricted to be below 30. Some spline based DQMs have been proposed to avoid this problem, but the construction of these schemes depends strongly on the degree of the considered B-spline (see for instance [
2
] and [
5
]). In this work we present a general DQM based on interpolation and quasi-interpolation. Firstly, we consider the construction of compactly supported cardinal functions
L
that interpolate the Kronecker sequence. They are linear combinations of translates of a B-spline
M
n
centered at the origin. Then, we revise some spline discrete quasi-interpolants defined from the same B-splines. We are interested in some recently defined and analyzed quasi-interpolants (cf. [
3
]). They are constructed by minimizing an error constant appearing in a particular expression of the quasi-interpolation error for regular enough functions. Finally, both the interpolants and the quasi-interpolants are used to define new interpolants having compactly supported fundamental functions again, and the maximal order of approximation, and the quintic case is described and compared with the results obtained in [
5
].