A new numerical method based on Bernstein polynomials expansion is proposed for solving one-dimensional elliptic interface problems. Both Galerkin formulation and collocation formulation are constructed to determine the expansion coefficients. In Galerkin formulation, the flux jump condition can be imposed by the weak formulation naturally. In collocation formulation, the results obtained by B-polynomials expansion are compared with that obtained by Lagrange basis expansion. Numerical experiments show that B-polynomials expansion is superior to Lagrange expansion in both condition number and accuracy. Both methods can yield high accuracy even with small value of N.
Jiankang Liu. Zhoushun Zheng. Qinwu Xu. "Bernstein-Polynomials-Based Highly Accurate Methods for One-Dimensional Interface Problems." J. Appl. Math. 2012 1 - 11, 2012. https://doi.org/10.1155/2012/859315