[Paper Review] A Bernstein Polynomial Collocation Method for the Solution of Elliptic Boundary Value Problems

Nikola Mirkov, Boško Rašuo|arXiv (Cornell University)|Nov 15, 2012

Numerical methods in engineering10 references3 citations

TL;DR

This paper introduces a novel Bernstein polynomial collocation method for solving elliptic boundary value problems, using global expansions in tensor-product Bernstein polynomials over arbitrary intervals [a,b]. The method achieves high accuracy and efficiency by leveraging the nodal value property of Bernstein coefficients, enabling direct boundary condition imposition and producing sparse linear systems, with demonstrated convergence and accuracy on Poisson, Helmholtz, and biharmonic equations using open-source code and a custom bernstein-poly library.

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.

Motivation & Objective

• To develop a high-order, efficient numerical method for solving 2D elliptic boundary value problems using Bernstein polynomials.
• To exploit the nodal value property of Bernstein basis functions to simplify direct imposition of Dirichlet and Neumann boundary conditions.
• To formulate a point-collocation method that mimics pseudospectral efficiency while avoiding orthogonality constraints of traditional spectral methods.
• To implement and validate the method on benchmark PDEs (Poisson, Helmholtz, biharmonic) with analytical solutions.
• To release an open-source Python library, bernstein-poly, for extensible use in solving complex PDEs.

Proposed method

• Approximate the solution of elliptic PDEs using a global expansion in tensor-product Bernstein polynomials over a rectangular domain [a,b]×[c,d].
• Use the property that expansion coefficients directly represent nodal values at the basis function nodes, enabling direct enforcement of boundary conditions at domain boundaries.
• Formulate a point-collocation method where the differential operator is applied at selected collocation points, leading to a sparse linear system.
• Compute derivatives of Bernstein basis functions using a closed-form expression involving binomial coefficients and shifted indices over arbitrary intervals.
• Construct the system matrix and right-hand side vector by evaluating the differential operator and boundary conditions at collocation points, using a structured grid.
• Implement the method in a modular, extensible open-source Python library, bernstein-poly, supporting both Dirichlet and Neumann boundary conditions.

Experimental results

Research questions

• RQ1 Can a collocation method based on global Bernstein polynomial expansions achieve high accuracy and efficiency for 2D elliptic PDEs?
• RQ2 How does the nodal value property of Bernstein polynomials simplify the imposition of Dirichlet and Neumann boundary conditions?
• RQ3 What is the convergence behavior and error performance of the method when applied to Poisson, Helmholtz, and biharmonic equations?
• RQ4 Can the method be efficiently implemented in a sparse matrix framework suitable for large-scale problems?
• RQ5 To what extent can the proposed open-source bernstein-poly library be extended to solve complex PDEs, including Navier-Stokes equations?

Key findings

• The method achieves high accuracy and spectral-like convergence rates when solving Poisson, Helmholtz, and biharmonic equations with both Dirichlet and Neumann boundary conditions.
• The direct imposition of boundary conditions via nodal values simplifies implementation and avoids the need for penalty or augmentation techniques.
• The resulting linear systems are sparse, with a structure typical of tensor-product grid discretizations, enabling efficient solution with standard solvers.
• The method demonstrates robust convergence and low error when compared to analytical solutions, particularly at higher polynomial orders (e.g., n=12 in the Poisson example).
• The open-source bernstein-poly library enables rapid prototyping and extension of the method to more complex PDEs, with minimal code required for new problems.
• The approach is shown to be effective and efficient, with a small number of lines of code needed to set up and solve non-trivial 2D boundary value problems.