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

Bin Han, Michelle Michelle

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 $Γ$, 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 $Γ$. 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 $d\ge 2$.

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

Abstract

The solution of an elliptic interface problem in a domain is often smooth away from the interface , but its gradient is discontinuous across , resulting in low regularity; in particular, . 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 th approximation order for any integer . Additionally, we rigorously prove that its convergence rates are of order in the -norm and order in the -norm, which are optimal with respect to the scheme's approximation order . Our approach involves incorporating wavelet basis functions from higher scale levels to capture the singularity in the neighbourhood of the interface . 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 -dimensional elliptic interface problems with .
Paper Structure (4 sections, 2 theorems, 53 equations, 3 tables)

This paper contains 4 sections, 2 theorems, 53 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Main Result Using Biorthogonal Multiwavelets on Bounded Intervals
  3. Numerical Experiments
  4. Proof of \ref{['thm:main']}

Key Result

Theorem 1

Let $m\geqslant 2$ be an integer and $u \in H^{1}_0(\Omega)$ be the true solution of the model problem model with variable functions $a,f$ such that $u_+ \in H^{m}(\Omega_+)$ and $u_- \in H^{m}(\Omega_-)$. Define $V^{wav}_{h} := \text{span}(\mathcal{B}^{S,H^{1}_0(\Omega)}_{J_0,J})$, where $\mathcal{ and the basic compactly supported boundary dual wavelets $\psi^L$ on $[0,\infty)$ and $\psi^R$ on $

Theorems & Definitions (7)

  • Theorem 1
  • Example 1
  • Example 2
  • Example 3
  • Proposition 2
  • proof
  • proof : Proof of \ref{['thm:main']}