Position Of Collocation Points Research Articles

A Robust Spline Collocation Method for Pricing American Put Options

In this paper a robust numerical method is proposed for pricing American put options. The Black-Scholes differential operator in the original form is discretized by using a quadratic spline collocation method on a piecewise uniform mesh for the spatial discretization and the implicit Euler scheme for the time discretization. The position of collocation points is chosen so that the spline difference operator satisfies the discrete maximum principle, which guarantees that the scheme is maximum-norm stable. The error estimation is derived by applying the maximum principle to the discrete linear complementarity problem in two mesh sets. It is proved that the scheme is second-order convergent with respect to the spatial variable and first-order convergent with respect to the time variable. Numerical results demonstrate that the scheme is stable and accurate.

CAD-based collocation eigenanalysis of 2-D elastic structures

This paper investigates the performance of the global collocation method for the numerical eigenfrequency extraction of 2-D elastic structures. The method is applied to CAD-based macroelements, starting from the older blending function Coons-Gordon interpolation (based on Lagrange polynomials) and extending to tensor product Bézier and B-splines. Numerical findings show equivalence between Lagrangian and Bézierian macroelements, while a mass lumping procedure is proposed for the former ones. Concerning B-splines, the influence of multiplicity of inner knots and the position of collocation points is thoroughly investigated. The theory is supported by 2-D numerical examples on rectangular and curvilinear structures of simple shape under plane stress conditions, in which the approximate solution rapidly converges towards the exact solution faster than that of the conventional finite element of similar mesh density.

A robust layer-resolving spline collocation method for a convection–diffusion problem

We consider finite difference approximation of a singularly perturbed one-dimensional convection–diffusion two-point boundary value problem. The problem is numerically treated by a quadratic spline collocation method on a piecewise uniform slightly modified Shishkin mesh. The position of collocation points is chosen so that the obtained scheme satisfies the discrete minimum principle. We prove pointwise convergence of order O ( N - 2 ln 2 N ) inside the boundary layer and second order convergence elsewhere. The uniform convergence of the approximate continual solution is also given. Further, we approximate normalized flux and give estimates of the error at the mesh points and between them. The numerical experiments presented in the paper confirm our theoretical results.

SENSITIVITY ANALYSIS FOR BOUNDARY ELEMENT ERROR ESTIMATION AND MESH REFINEMENT

The subject of this paper is the sensitivity analysis of approximate boundary element solutions with respect to the positions of the collocation points. The direct differentiation approach is considered here and the analysis is performed analytically. Since only the collocation points are perturbed, the shape of the body and the corresponding discretization remain unaltered. This aspect makes the present work quite different in spirit with respect to earlier analyses on shape sensitivities. Sensitivities of approximate BEM solutions with respect to the positions of collocation points are shown to be related to the residual of hypersingular integral equations. Numerical results confirm that the present approach can be seen as the analytical counterpart of an adaptive scheme for mesh refinement presented by the same author in some recent papers. Some other advantages of the present approach over the former one are also outlined.

Collocation in finite element analysis of constrained problems

The concept of three-field displacement finite element formulations of constrained problems is explored in this paper. The approach studied uses collocation within the element domain to enforce the constraints of the problem and resembles mixed formulations because in addition to the approximation of the displacement field, all other auxiliary deformation variables that enter the constraint conditions are interpolated independently. Plate-bending as described by the Reissner–Mindlin theory is used as a model problem for the development of the approach. Two triangular plate elements are formulated based on this methodology, and their performance is investigated using several patch tests. Numerical examples are also considered to study the element behavior under locking conditions (which occur with the classical displacement approach at the thin plate limit). The ability of the elements to overcome locking is established using mathematical arguments and practical examples. The influence of the position of collocation points on the computed results is evaluated through sensitivity studies, with the aim to identify the optimal set. Results are compared with those obtained from the exact solutions and the associated classical displacement model with selective reduced integration of the constrained energy terms. Key words: finite elements, collocation, discrete constraints, mixed methods, thick plates, bending of plates, shear deformations.