A Bernstein Polynomial Collocation Method for the Solution of Elliptic Boundary Value Problems

Abstract

In this article, a formulation of a point-collocation method in which the unknown function is approximated using global expansion in tensor product Bernstein polynomial basis is presented. Bernstein polynomials used in this study are defined over general interval [a,b]. Method incorporates several ideas that enable higher numerical efficiency compared to Bernstein polynomial methods that have been previously presented. The approach is illustrated by a solution of Poisson, Helmholtz and Biharmonic equations with Dirichlet and Neumann type boundary conditions. Comparisons with analytical solutions are given to demonstrate the accuracy and convergence properties of the current procedure. The method is implemented in an open-source code, and a library for manipulation of Bernstein polynomials bernstein-poly, developed by the authors.