A semilinear singularly perturbed reaction-diffusion problem is considered and the approximate solution is given in the form of a quadratic polynomial spline. Using the collocation method on a simple piecewise equidistant mesh, an approximation almost second order uniformly accurate in small parameter is obtained. Numerical results are presented in support of this result. AMS Mathematics Subjec...

Two methods are presented for approximating the costate of optimal control problems in integral form using orthogonal collocation at Legendre–Gauss (LG) and Legendre–Gauss–Radau (LGR) points. It is shown that the derivative of the costate of the continuous-time optimal control problem is equal to the negative of the costate of the integral form of the continuous-time optimal control problem. Us...

The Karhunen-Loève expansion of a Gaussian process, a common tool on finite element methods for differential equations with stochastic coefficients, is based on the spectral decomposition of its covariance function. The eigenpairs of the covariance are expressed as a Fredholm integral equation of second kind, which can be readily approximated with piecewise-constant finite elements. In this wor...

We consider the problem of scattering of a time-harmonic acoustic incident plane wave by a sound soft convex polygon. Standard boundary or finite element methods, with a piecewise polynomial approximation space, have a computational cost that grows linearly with respect to the frequency of the incident wave. Recently ChandlerWilde and Langdon proposed a novel Galerkin boundary element method fo...

Symmetric collocation, which can be used to numerically solve linear partial differential equations, is a natural generalization of the well-established scattered data interpolation method known as radial basis function (rbf) interpolation. As with rbf interpolation, a major shortcoming of symmetric collocation is the high cost, in terms of floating point operations, of evaluating the obtained ...

In this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with natural logarithm piecewise in the conventional governing equations. We converted into an using source points collocated outside domain to ensure that distance was always greater than zero avoid singularity...

Time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations (RPIEs). We outline how Fourier and Laplace transforms can be used to obtain an O(h) convergence result for a “polar” piecewise–linear collocation approximation of a scalar RPIE on an infinite flat plate.

An overlapping domain decomposition, additive Schwarz, conjugate gradient method is presented for the solution of the linear systems which arise when orthogonal spline collocation with piecewise Hermite bicu-bics is applied to the Dirichlet problem for Poisson's equation on a rectangle .

We consider the numerical solution of the problem of time-harmonic acoustic scattering in two dimensions by a sound soft convex polygon. Standard boundary or finite element methods with piecewise polynomial approximation spaces have a computational cost that grows at least linearly with respect to the frequency of the incident wave. By including in the approximation space the products of plane ...

We propose and analyze a new spectral collocation method to solve eigenvalue problems of compact integral operators, particularly, piecewise smooth operator kernels and weakly singular operator kernels of the form 1 |t−s|μ , 0 < μ < 1. We prove that the convergence rate of eigenvalue approximation depends upon the smoothness of the corresponding eigenfunctions for piecewise smooth kernels. On t...