On the Asymptotic Convergence of Spline Collocation Methods for Partial Differential Equations

Abstract

We examine the asymptotic accuracy of the method of collocation for the approximate solution of linear elliptic partial differential equations. Specifically we consider the nodal collocation of a second order equation in the plane with biperiodicity conditions using tensor product smooth splines of odd degree as trial functions. We prove optimal rates of convergence in L for partial derivatives of the approximate solution which are of order at least two in one variable, while the solution itself and its gradient converge in L at rates less than the optimal approximation theoretic results.

Linear elastic fracture simulation directly from CAD: 2D NURBS-based implementation and role of tip enrichment

“…This is in consistence with the basic feature of the BEM since only the unknown elds (displacement and traction) along the boundary is required to approximate. And the convergence of collocation BEM with splines has been investigated which forms a solid basis for the combined methodology [69][70] and latest work can be referred in [71]. In this paper, a new application of IGABEM is discussed in detail for linear elastic fracture problems.…”

Isogeometric Collocation Methods

“…Other approaches for the theoretical analysis of the collocation method with spline spaces in higher dimensions are given, for particular cases, in Arnold et al 3 and Prenter. …”

On the Spline Collocation Method for the Single-Layer Heat Operator Equation

“…The collocation equations corresponding to the equation Su = f are: Find uA£Ji such that (3.1) (SuA)(en,tm) = f(en,tm), \<n<N, \<m<M. We adapt the method developed by Arnold and Wendland [3] for the onedimensional case and extended by Arnold and Saranen [2] for biperiodic problems. In this approach the collocation problem is reduced to a Galerkin problem.…”