Are best approximations really better than Chebyshev?

Best and Chebyshev approximations play an important role in approximation theory. From the viewpoint of measuring approximation error in the maximum norm, it is evident that best approximations are better than their Chebyshev counterparts. However, the situation may be reversed if we compare the approximation quality from the viewpoint of either the rate of pointwise convergence or the accuracy of spectral differentiation. We show that when the underlying function has an algebraic singularity, the Chebyshev projection of degree n converges one power of n faster than its best counterpart at each point away from the singularity and both converge at the same rate at the singularity. This gives a complete explanation for the phenomenon that the accuracy of Chebyshev projections is much better than that of best approximations except in a small neighborhood of the singularity. Extensions to superconvergence points and spectral differentiation, Chebyshev interpolants and other orthogonal projections are also discussed.Comment: 23 page

Error Analysis of Semidiscrete Finite Element Methods for Inhomogeneous Time-Fractional Diffusion

We consider the initial boundary value problem for the inhomogeneous time-fractional diffusion equation with a homogeneous Dirichlet boundary condition and a nonsmooth right hand side data in a bounded convex polyhedral domain. We analyze two semidiscrete schemes based on the standard Galerkin and lumped mass finite element methods. Almost optimal error estimates are obtained for right hand side data $f(x,t)\in L^\infty(0,T;\dot H^q(\Omega))$, $-1< q \le 1$, for both semidiscrete schemes. For lumped mass method, the optimal $L^2(\Omega)$-norm error estimate requires symmetric meshes. Finally, numerical experiments for one- and two-dimensional examples are presented to verify our theoretical results.Comment: 21 pages, 4 figure