A Posteriori Error Estimates of Spectral Approximations for Second Order Partial Differential Equations in Spherical Geometries

In this paper, we investigate a posteriori error estimates of the Galerkin spectral methods for second-order equations, and propose a simple type of error estimator comprising expansion coefficients of known quantities such as the right-hand term. We first show that the errors of the numerical solution of the Poisson equation on the unit ball in arbitrary dimensions can be identified by the approximation errors of the (weighted) $$L^2$$ L 2 -projection of the right-hand function together with the non-homogeneous boundary function. This result indicates that the decay rate of the high frequency coefficients of the right-hand term in weighted orthogonal ball polynomials and of the boundary term in spherical harmonics serves as an ideal a posteriori error estimator. In the sequel, we establish a posteriori error estimates on the Galerkin spectral method applied to the singular perturbation problem of a reaction–diffusion equation on the unit ball. Again, the efficiency is given by the approximation errors of the weighted $$L^2$$ L 2 -projection of the right-hand function; while the reliability is determined by the truncation errors of the right-hand function together with exponentially decaying multiples of the low frequency coefficients, which also reveals that the a posterior error estimator is dominated by the decay rate of the high frequency coefficients of the right-hand term. Finally, numerical examples are presented to illustrate the theoretical results.