We consider Quadratic Spline Collocation (QSC) methods for linear second order elliptic Partial Differential Equations (PDEs). The standard formulation of these methods leads to non-optimal approximations. In order to derive optimal QSC approximations, high order perturbations of the PDE problem are generated. These perturbations can be applied either to the PDE problem operators or to the right sides, thus leading to two different formulations of optimal QSC methods. The convergence properties of the QSC methods are studied. OptimalO(h
3−j
) global error estimates for thejth partial derivative are obtained for a certain class of problems. Moreover,O(h
4−j
) error bounds for thejth partial derivative are obtained at certain sets of points. Results from numerical experiments verify the theoretical behaviour of the QSC methods. Performance results also show that the QSC methods are very effective from the computational point of view. They have been implemented efficiently on parallel machines.

/pdf/quadratic-spline-collocation-methods-for-elliptic-partial-1kqzuf9jil.pdf

Quadratic spline collocation methods for elliptic partial differential equations

Redefined cubic B-splines collocation method for solving convection–diffusion equations

Cubic splines collocation methods for unilateral problems

In this paper, we use uniform cubic spline polynomials to derive some new consistency relations These relations are then used to develop a numerical method for computing smooth approximations to the solution and its first, second as well as third derivatives for a second order boundary value problem The present method outperforms other collocations, finite-difference and splines methods of the same order Numerical illustrations are provided to demonstrate the practical use of our method

Cubic spline method for solving two-point boundary-value problems

https://www.tpbin.com/Uploads/Subjects/7890b0be-c060-4018-aabf-9388d039259b.pdf

Troesch’s problem: A B-spline collocation approach

: The report is concerned with the use of collocation by splines to numerically solve two-point boundary-value problems. The problem is analyzed in terms of cubic splines first and then extended to the use of quintic and septic splines. Consideration is given both to convergences as the mesh is refined and to the bandwidth of the matrices involved. Comparisons are made to a similar approach using the Galerkin method rather than collocation. (Author)

/pdf/a-collocation-method-for-two-point-boundary-value-problems-18jp46ky9w.pdf

A Collocation Method for Two-Point Boundary Value Problems,

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A Collocation Method for Two-Point Boundary Value Problems,