Poly-Sinc Solution of Stochastic Elliptic Differential Equations

Abstract

In this paper, we introduce a numerical solution of a stochastic partial differential equation (SPDE) of elliptic type using polynomial chaos along side with polynomial approximation at Sinc points. These Sinc points are defined by a conformal map and when mixed with the polynomial interpolation, it yields an accurate approximation. The first step to solve SPDE is to use stochastic Galerkin method in conjunction with polynomial chaos, which implies a system of deterministic partial differential equations to be solved. The main difficulty is the higher dimensionality of the resulting system of partial differential equations. The idea here is to solve this system using a small number of collocation points in space. This collocation technique is called Poly-Sinc and is used for the first time to solve high-dimensional systems of partial differential equations. Two examples are presented, mainly using Legendre polynomials for stochastic variables. These examples illustrate that we require to sample at few points to get a representation of a model that is sufficiently accurate.

Adaptive Piecewise Poly-Sinc Methods for Ordinary Differential Equations

“…A novel family of polynomial approximation called Poly-Sinc interpolation which interpolate data of the form {x k , y k } N k=−M where {x k } N k=−M are Sinc points, were derived in [23,30] and extended in [24]. The interpolation to this type of data is accurate provided that the function y with values y k = y(x k ) belong to the space of analytic functions [30,31]. For the ease of presentation and discussion, we assume that M = N. Poly-Sinc approximation was developed in order to mitigate the poor accuracy associated with differentiating the Sinc approximation when approximating the derivative of functions [23].…”

“…Theoretical frameworks on the error analysis of function approximation, quadrature, and the stability of the Poly-Sinc approximation were studied in [23,24,32,33]. Furthermore, Poly-Sinc approximation was used to solve BVPs in ordinary and partial differential equations [31,[34][35][36][37][38]. We start with a brief overview of Lagrange interpolation.…”

Adaptive Piecewise Poly-Sinc Methods for Ordinary Differential Equations

“…Theoretical frameworks on the error analysis of function approximation, quadrature, and the stability of the Poly-Sinc approximation were studied in [23,24,32,33]. Furthermore, Poly-Sinc approximation was used to solve BVPs in ordinary and partial differential equations [31,[34][35][36][37][38]. We start with a brief overview of Lagrange interpolation.…”

Weighted Lagrange Interpolation Using Orthogonal Polynomials: Stenger's Conjecture, Numerical Approach

“…As a study interval we use we choose [a, b] = [−1, 1]. It is known that Poly-Sinc shows high accuracy even with used number of Sinc points [13,14]. So, we will test the eigenvalues for B ± for not so huge numbers, roughly we test up to 41 Sinc points.…”