New solvers for higher dimensional poisson equations by reduced B‐splines

Abstract

We use higher dimensional B-splines as basis functions to find the approximations for the Dirichlet problem of the Poisson equation in dimension two and three. We utilize the boundary data to remove unnecessary bases. Our method is applicable to more general linear partial differential equations. We provide new basis functions which do not require as many B-splines. The number of new bases coincides with that of the necessary knots. The reducing process uses the boundary conditions to redefine a basis without extra artificial assumptions on knots which are outside the domain. Therefore, more accuracy would be expected from our method. The approximation solutions satisfy the Poisson equation at each mesh point and are solved explicitly using tensor product of matrices.