http://home.eps.hw.ac.uk/~ab226/tmp/implicit/lele_92.pdf

Compact finite difference schemes with spectral-like resolution

Numerical Techniques in Electromagnetics is designed to show the reader how to pose, numerically analyze, and solve electromagnetic (EM) problems. It gives them the ability to expand their problem-solving skills using a variety of available numerical methods. Topics covered include fundamental concepts in EM; numerical methods; finite difference methods; variational methods, including moment methods and finite element methods; transmission-line matrix or modeling (TLM); and Monte Carlo methods. The simplicity of presentation of topics throughout the book makes this an ideal text for teaching or self-study by senior undergraduates, graduate students, and practicing engineers.

Numerical Techniques in Electromagnetics, Second Edition

Approximations to an isolated solution of an mth order nonlinear ordinary differential equation with m linear side conditions are determined. These approximations are piecewise polynomial functions of order $m + k$ (degree less than $m + k$) possessing $m - 1$ continuous derivatives. They are determined by collocation, i.e., by the requirement that they satisfy the differential equation at k points in each subinterval, together with the m side conditions. If the solution of the sufficiently smooth differential equation problem has $m + 2k$ continuous derivatives and if the collocation points are the zeroes of the kth Legendre polynomial relative to each subinterval, then the global error in these approximations is $O(| \Delta |^{m + k} )$ with $| \Delta |$ the maximum subinterval length. Moreover, at the ends of each subinterval, the approximation and its first $m - 1$ derivatives are $O(| \Delta |^{2k} )$ accurate. The solution of the nonlinear collocation equations may itself be approximated by solving ...

Collocation at Gaussian Points

Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations

PDECOL, new computer software package for numerically solving coupled systems of nonlinear partial differential equations (PDE's) in one space and one time dimension, is discussed. The package implements finite element collocation methods based on piecewise polynomials for the spatial discretization techniques. The time integration process is then accomplished by widely acceptable procedures that are generalizations of the usual methods for treating time-dependent partial differental equations. PDECOL is unique because of its flexibiility both in the class of problems it addresses and in the variety of methods it provides for use in the solution process. High-order methods (as well as low-order ones) are readily available for use in both the spatial and time discretization procedures. The time integration methods used feature automatic time step size and integration formula order selection so as to solve efficiently the problem at hand and yet achieve a user-specific time integration error level. PDECOL consists of a collection of 19 subroutines written in reasonably standard Fortran, and therefore is quite portable. No special hardware features are required. PDECOL is designed to solve broad classes of difficult systems of partial differential equations that descrobe physical processes. 4 figures, 1 table. (RWR)

Algorithm 540: PDECOL, General Collocation Software for Partial Differential Equations [D3]

Error bounds for spline and L-spline interpolation

The finite element method, using smooth splines as basis functions, applied to the model problem $u_t = cu_x $ with periodic data generates a differential-difference equation whose phase error is closely estimated and compared with the phase error of both explicit and high order implicit centered differencing. We also compute and compare the minimum work required to obtain a fixed error for several fully discrete schemes.

The Relative Efficiency of Finite Difference and Finite Element Methods. I: Hyperbolic Problems and Splines

In this short note we consider the periodic initial value problem for the semibounded differential equation $u_t = \sum _{k = 0}^p {a_k ({\partial / {\partial x}})^k u} $, where the $a_k $ are constants. We observe that the spline–Galerkin scheme arising from splines of order $\mu $, analyzed by Thomee, has precisely the same solution as the collocation scheme using a basis of cardinal splines of order $2\mu $. The fact that both methods have the convergence rate $O(h^{2(\mu - [{p / {2]}})} )$ then follows from known error bounds for spline interpolation and the stability of the Galerkin schemes.

https://www.jstor.org/stable/pdfplus/2156038.pdf

The Relation Between the Galerkin and Collocation Methods Using Smooth Splines

In the previous paper by A Meir and A Sharma, error bounds for lacunary interpolation of certain functions by deficient quintic splines are developed In this note, we extend their results to a wider class of functions and indicate that the extended results are best possible In addition, a stability result for such interpolation is also presented

A Note on Lacunary Interpolation by Splines

/pdf/the-comparative-efficiency-of-certain-finite-element-and-3iwka521pg.pdf

Blair Swartz

Papers

Error bounds for spline and L-spline interpolation

The Relative Efficiency of Finite Difference and Finite Element Methods. I: Hyperbolic Problems and Splines

The Relation Between the Galerkin and Collocation Methods Using Smooth Splines

A Note on Lacunary Interpolation by Splines

The comparative efficiency of certain finite element and finite difference methods for a hyperbolic problem