1994
DOI: 10.1007/bf01935015
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Quadratic spline collocation methods for elliptic partial differential equations

Abstract: Summary. We consider Quadratic Spline Collocation (QSC) methods for linear second order elliptic Partial Differential Equations (PDEs). The standard Cannulation of these methods leads to non-optimal approximations. In order (0 derive optimal QSC approximations, high order perturbations of the PDE problem arc generated. These perturbations can be applied either to die POE problem operators or to the right sides, thus leading 10 two different fonnulations of optimal QSC methods. The convergence properties of the… Show more

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Cited by 54 publications
(25 citation statements)
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“…where p 2 is a positive constant independent of h. Using the Schwartz inequality and the inverse estimate (see [6]), we have a positive constant c independent of h such that (7.5b)…”
Section: General Preconditioningmentioning
confidence: 97%
See 2 more Smart Citations
“…where p 2 is a positive constant independent of h. Using the Schwartz inequality and the inverse estimate (see [6]), we have a positive constant c independent of h such that (7.5b)…”
Section: General Preconditioningmentioning
confidence: 97%
“…The case using cubic splines and Gaussian points was analyzed in [14] and similar problems were discussed in [5], [16] and [18] for the case where there is only one finite element space and A NiM and fi NM are finite element discretizations. The collocation method using quadratic interpolatory splines was used for the numerical solution of two point boundary value problems in [15] and for linear second order elliptic partial differential equations in [6]. Some solvers for quadratic spline collocation equations were developed in [7].…”
Section: ~ (P N M U U) H -mentioning
confidence: 99%
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“…Here N is the distribution matrix for the permittivity (n 2 ), and the coefficient matrix for the zero-order, the first-order, and the secondorder derivatives are given by [17]:…”
Section: Structure and Working Principlementioning
confidence: 99%
“…The original second order differential equation is therefore transferred to a set of algebraic equations, which can be solved by matrix techniques. Spline collocation method is considered as one of finite element methods (FEM) [17], as it utilizes finite element, namely, localized basis functions which can become infinitely small when the grid size of the computation window is decreased. However, the spline collocation method uses the weights of the basis-functions as the undetermined values and forces the approximation functions to exactly match the unknown functions at a set of points.…”
Section: Introductionmentioning
confidence: 99%