Müntz--Galerkin Methods and Applications to Mixed Dirichlet--Neumann Boundary Value Problems

Abstract

Solutions for many problems of interest exhibit singular behaviors at domain corners or points where the boundary condition changes type. For these types of problems, direct spectral methods with the usual polynomial basis functions do not lead to a satisfactory convergence rate. We develop in this paper a Müntz--Galerkin method which is based on specially tuned Müntz polynomials to deal with the singular behaviors of the underlying problems. By exploring the relations between Jacobi polynomials and Müntz polynomials, we develop efficient implementation procedures for the Müntz--Galerkin method, and provide optimal error estimates. As an example of applications, we consider the Poisson equation with mixed Dirichlet--Neumann boundary conditions, whose solution behaves like $O(r^{1/2})$ near the singular point, and demonstrate that the Müntz--Galerkin method greatly improves the rates of convergence of the usual spectral method.