A New Collocation Scheme Using Non-polynomial Basis Functions
Date of Issue2017
School of Physical and Mathematical Sciences
In this paper, we construct a set of non-polynomial basis functions from a generalised Birkhoff interpolation problem involving the operator: Lλ=d2/dx2−λ2Lλ=d2/dx2−λ2 with constant λ.λ. With a direct inverting the operator, the basis can be pre-computed in a fast and stable manner. This leads to new collocation schemes for general second-order boundary value problems with (i) the matrix corresponding to the operator LλLλ being identity; (ii) well-conditioned linear systems and (iii) exact imposition of various boundary conditions. This also provides efficient solvers for time-dependent nonlinear problems. Moreover, we can show that the new basis has the approximability to general functions in Sobolev spaces as good as orthogonal polynomials.
Generalised Birkhoff Interpolation Problem
Journal of Scientific Computing
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