A Derivative-Orthogonal Wavelet Multiscale Method for 1D Elliptic Equations with Rough Diffusion Coefficients

TL;DR

This paper introduces a derivative-orthogonal wavelet multiscale method for 1D elliptic equations with rough, high-contrast coefficients, achieving robust, accurate solutions on coarse meshes with proven error bounds and stable condition numbers.

Contribution

The paper develops a novel wavelet-based framework that ensures condition number bounds independent of mesh size and provides first- and second-order error convergence for rough coefficients.

Findings

Condition number bounded by the ratio of max to min coefficient values

First-order energy norm and second-order L2 norm error convergence

Effective handling of high-frequency, high-contrast coefficient oscillations

Abstract

In this paper, we investigate 1D elliptic equations $-\nabla\cdot (a\nabla u)=f$ with rough diffusion coefficients $a$ that satisfy $0<a_{\min}\le a\le a_{\max}<\infty$ and $f\in L_2(\Omega)$. To achieve an accurate and robust numerical solution on a coarse mesh of size $H$, we introduce a derivative-orthogonal wavelet-based framework. This approach incorporates both regular and specialized basis functions constructed through a novel technique, defining a basis function space that enables effective approximation. We develop a derivative-orthogonal wavelet multiscale method tailored for this framework, proving that the condition number $\kappa$ of the stiffness matrix satisfies $\kappa\le a_{\max}/a_{\min}$, independent of $H$. For the error analysis, we establish that the energy and $L_2$-norm errors of our method converge at first-order and second-order rates, respectively, for any coarse mesh $H$. Specifically, the energy and $L_2$-norm errors are bounded by $2 a_{\min}^{-1/2} \|f\|_{L_2(\Omega)} H$ and $4 a_{\min}^{-1}\|f\|_{L_2(\Omega)} H^2$. Moreover, the numerical approximated solution also possesses the interpolation property at all grid points. We present a range of challenging test cases with continuous, discontinuous, high-frequency, and high-contrast coefficients $a$ to evaluate errors in $u, u'$ and $a u'$ in both $l_2$ and $l_\infty$ norms. We also provide a numerical example that both coefficient $a$ and source term $f$ contain discontinuous, high-frequency and high-contrast oscillations. Additionally, we compare our method with the standard second-order finite element method to assess error behaviors and condition numbers when the mesh is not fine enough to resolve coefficient oscillations. Numerical results confirm the bounded condition numbers and convergence rates, affirming the effectiveness of our approach.