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.

翻译：本文研究一维椭圆方程 $-\nabla\cdot (a\nabla u)=f$，其中扩散系数 $a$ 具有粗糙性，满足 $0<a_{\min}\le a\le a_{\max}<\infty$ 且 $f\in L_2(\Omega)$。为在尺寸为 $H$ 的粗网格上获得精确且稳健的数值解，我们提出一种基于导数正交小波的框架。该方法结合了通过一种新颖技术构造的常规基函数与特殊基函数，定义了一个能够实现有效逼近的基函数空间。我们为此框架专门设计了一种导数正交小波多尺度方法，并证明刚度矩阵的条件数 $\kappa$ 满足 $\kappa\le a_{\max}/a_{\min}$，且与 $H$ 无关。在误差分析方面，我们证明了对于任意粗网格 $H$，该方法的能量范数误差和 $L_2$ 范数误差分别以一阶和二阶速率收敛。具体而言，能量范数误差和 $L_2$ 范数误差分别以 $2 a_{\min}^{-1/2} \|f\|_{L_2(\Omega)} H$ 和 $4 a_{\min}^{-1}\|f\|_{L_2(\Omega)} H^2$ 为界。此外，数值近似解在所有网格点上也具有插值性质。我们提出了一系列具有挑战性的测试案例，包括连续、间断、高频及高对比度的系数 $a$，以评估 $u$、$u'$ 和 $a u'$ 在 $l_2$ 和 $l_\infty$ 范数下的误差。我们还提供了一个数值示例，其中系数 $a$ 和源项 $f$ 均包含间断、高频和高对比度的振荡。此外，我们将本方法与标准的二阶有限元方法进行比较，以评估当网格不够精细以解析系数振荡时的误差行为和条件数。数值结果证实了条件数有界性和收敛速率，验证了本方法的有效性。