Asymptotic Coefficients and Errors for Chebyshev Polynomial Approximations with Weak Endpoint Singularities: Effects of Different Bases

When solving differential equations by a spectral method, it is often convenient to shift from Chebyshev polynomials $T_n(x)$ with coefficients $a_n$ to modified basis functions that incorporate the boundary conditions. For homogeneous Dirichlet boundary conditions, $u(\pm 1)=0$, popular choices include the ``Chebyshev difference basis", $\varsigma_{n}(x) \equiv T_{n+2}(x)

• T_{n}(x)$withcoefficientsheredenoted$b_{n}$ and the ``quadratic-factor basis functions" ${\varrho }_n(x)\equiv (1-x^2)T_n(x)$ with coefficients $c_n$. If $u(x)$ is weakly singular at the boundaries, then $a_n$ will decrease proportionally to $\mathcal{O}(A(n)\mathrm{/}{n}^{\kappa })$ for some positive constant $\kappa$, where the $A(n)$ is a logarithm or a constant. We prove that the Chebyshev difference coefficients $b_n$ decrease more slowly by a factor of $1\mathrm{/}n$ while the quadratic-factor coefficients $c_n$ decrease more slowly still as $\mathcal{O}(A(n)\mathrm{/}{n}^{\kappa -2})$. The error for the unconstrained Chebyshev series, truncated at degree $n=N$, is $\mathcal{O}(\mathrm{\mid }A(N)\mathrm{\mid }\mathrm{/}{N}^{\kappa })$ in the interior, but is worse by one power of $N$ in narrow boundary layers near each of the endpoints. Despite having nearly identical error \emph{norms}, the error in the Chebyshev basis is concentrated in boundary layers near both endpoints, whereas the error in the quadratic-factor and difference basis sets is nearly uniform oscillations over the entire interval in $x$. Meanwhile, for Chebyshev polynomials and the quadratic-factor basis, the value of the derivatives at the endpoints is $\mathcal{O}(N^2)$, but only $\mathcal{O}(N)$ for the difference basis.