155 research outputs found
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 , , for both semidiscrete schemes. For lumped mass method, the optimal
-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
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