Wavelet-based Galerkin Scheme with Arbitrarily High-Order Convergence for 1D Elliptic Interface Problems

The solution $u$ of an elliptic interface problem in a domain $\Omega$ is often smooth away from the interface $\Gamma\subset \Omega$, but its gradient is discontinuous across $\Gamma$, resulting in low regularity; in particular, $u \notin H^{1.5}(\Omega)$. This paper focuses on 1D elliptic interface problems using wavelet methods. We propose a Galerkin method using locally supported biorthogonal wavelet bases on bounded intervals with $m$th approximation order for any integer $m \ge 2$. Additionally, we rigorously prove that its convergence rates are of order $m-1$ in the $H^1(\Omega)$-norm and order $m$ in the $L^2(\Omega)$-norm, which are optimal with respect to the scheme's approximation order $m$. Our approach involves incorporating wavelet basis functions from higher scale levels to capture the singularity in the neighbourhood of the interface $\Gamma$. The results in this paper both complement and sharply contrast our findings in Han and Michelle (2024), where we consider a similar wavelet-based method for solving $d$-dimensional elliptic interface problems with $d\ge 2$.