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

Abstract

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", $\varsigma_{n}(x) \equiv T_{n+2}(x) - T_{n}(x)$ with coefficients here denoted $b_{n}$ 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 $\mathcal{O}(A(n)/n^{\kappa})$ 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 $1/n$ while the quadratic-factor coefficients $c_{n}$ decrease more slowly still as $\mathcal{O}(A(n)/n^{\kappa-2})$. The error for the unconstrained Chebyshev series, truncated at degree $n=N$, is $\mathcal{O}(|A(N)|/N^{\kappa})$ 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.