Galerkin method

Convection–diffusion equation

Pure mathematics

Applied mathematics

Convergence (economics)

Dimension (graph theory)

Economics

Mathematical optimization

Solver

Thermodynamics

Spectral method

Physics

Finite element method

Advection

Diffusion

Mathematics

Mathematical analysis

Reaction–diffusion system

Economic growth

Discontinuous Galerkin method

Sobolev space

Fractional calculus

SIAM Journal on Numerical Analysis

We investigate a spectral Galerkin method for the fractional advection-diffusion-reaction equations in one dimension. We first prove sharp regularity estimates of solutions in nonweighted and weighted Sobolev spaces. Then we obtain optimal convergence orders of the spectral Galerkin methods for both fractional advection-diffusion and diffusion-reaction equations. We also present an iterative solver with a quasi-optimal complexity. Numerical results are presented to verify the theoretical analysis.

Optimal Regularity and Error Estimates of a Spectral Galerkin Method for Fractional Advection-Diffusion-Reaction Equations