Evaluation of spectral, spectral‐element and finite‐element methods for the solution of the pellet equation

Abstract

AbstractSeveral numerical methods (orthogonal collocation, Galerkin, tau, least‐squares and least‐squares with a direct minimization algorithm) are applied to solve a linear diffusion–reaction problem. The spectral, finite‐element and spectral‐element frameworks are employed to investigate the methods. Overall, the Galerkin and tau methods are considered the most universal methods.Spectral framework: With sufficient diffusion limitations, the least‐squares method suffers in general from significantly lower accuracy than the Galerkin, tau and orthogonal collocation methods. On the other hand, the least‐squares method with a direct minimization algorithm provides favourable lower system matrix condition numbers than the conventional least‐squares approach. Hence, for higher diffusion limitations, the least‐squares direct minimization formulation provides higher numerical accuracy than the conventional least‐squares method, but is still not as accurate as the Galerkin, tau and orthogonal collocation techniques. The accuracy of the least‐squares solution can compete with the other methods only in cases with low gradients in the solution.Element framework: For a highly diffusion limited problem, the element framework is considered favourable as compared to the spectral framework. On the other hand, the element approach is not as efficient as the spectral solution for small Thiele modulus solutions.