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

Abstract

The solution $u$ of an elliptic interface problem in a domain $Ω$ is often smooth away from the interface $Γ\subset Ω$, but its gradient is discontinuous across $Γ$. Consequently, $u$ has low global regularity and generally does not belong to $H^{3/2}(Ω)$. This paper studies 1D elliptic interface problems using wavelet methods. We propose a Galerkin method based on compactly supported biorthogonal wavelet bases on bounded intervals with approximation order $m$, for any integer $m \ge 2$. Our approach involves incorporating wavelet basis functions from higher scale levels to capture the singularity in the neighbourhood of the interface $Γ$. A principal contribution of this paper is a rigorous convergence analysis establishing the optimal orders $m-1$ in the $H^1(Ω)$-norm and $m$ in the $L^2(Ω)$-norm. The proof combines careful decay estimates for dual wavelet coefficients associated with wavelets supported away from the interface $Γ$ and with those supported in its neighborhood, wavelet characterizations of Sobolev spaces, and standard convergence arguments from the finite element method (FEM). Various numerical experiments are provided to verify the theoretical findings, including a special two-dimensional elliptic interface problem with vertical interface lines.