Theoretical analysis of pseudospectral convergence based on total variation and a related mesh refinement method

The pseudospectral method is often used to solve optimization problems, and its theoretical convergence is decisive to effectiveness and efficiency. A theoretical analysis of the pseudospectral convergence based on total variation is derived by extending differentiable functions' approximate convergence to multi-interval affine functions. The derived theoretical convergence shows good consistence with the convergence in numerical calculations. A novel mesh refinement algorithm is then proposed according to the theory. When developing the algorithm, a fixed polynomial order of the new intervals is assumed, and no increase in convergence error is taken as the primary criterion. Numerical simulations of two typical optimal problems are conducted to verify the convergence of the new algorithm. Its superiority is also inspected by comparing it with a ph adaptive algorithm. The results show that the proposed algorithm achieves a better convergence accuracy with fewer iterations and less computing time.