Wavelet-based Galerkin Scheme with Arbitrarily High-Order Convergence for 1D Elliptic Interface Problems
This provides a high-order numerical method for solving 1D elliptic interface problems, which is incremental as it extends wavelet techniques to 1D with rigorous proofs, complementing prior work on higher dimensions.
The paper tackles 1D elliptic interface problems, where solutions have discontinuous gradients across interfaces, by proposing a wavelet-based Galerkin method with locally supported biorthogonal bases, achieving optimal convergence rates of order m-1 in H^1-norm and m in L^2-norm for any integer m ≥ 2.
The solution $u$ of an elliptic interface problem in a domain $Ω$ is often smooth away from the interface $Î\subset Ω$, but its gradient is discontinuous across $Î$, resulting in low regularity; in particular, $u \notin H^{1.5}(Ω)$. 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 $H^1(Ω)$-norm and order $m$ in the $L^2(Ω)$-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 $Î$. 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 $d\ge 2$.