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

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 $f\in L_2(Ω)$. 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 $κ$ of the stiffness matrix satisfies $κ\le a_{\max}/a_{\min}$, 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 $2 a_{\min}^{-1/2} \|f\|_{L_2(Ω)} H$ and $4 a_{\min}^{-1}\|f\|_{L_2(Ω)} H^2$. 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.

Figures (11)

• Figure 1: Left panel: $\phi(2x-1)$. Middle and right panels: $\{\psi(2^jx-k):\ k=0,\ldots,2^j-1\}$ with $j=1$.
• Figure 2: $\{\psi(2^jx-k):\ k=0,\ldots,2^j-1\}$ with $j=2$.
• Figure 3: Left panel: $(\phi(2x-1))'$. Middle and right panels: $\{(\psi(2^jx-k))':\ k=0,\ldots,2^j-1\}$ with $j=1$.
• Figure 4: $\{(\psi(2^jx-k))':\ k=0,\ldots,2^j-1\}$ with $j=2$.
• Figure 5: $S_0$ (first panel) and $(S_0)'$ (second panel) in \ref{['Special:Si']} with $n=3$ in $[0,\frac{1}{4}]$ with a continuous coefficient function $a$. $S_0$ (third panel) and $(S_0)'$ (fourth panel) in \ref{['Special:Si']} with $n=3$ in $[0,\frac{1}{4}]$ with a discontinuous coefficient function $a$. Note that $\overline{\mathop{\mathrm{supp}}\nolimits(S_0)}=\overline{\mathop{\mathrm{supp}}\nolimits((S_0)')}=[0,\frac{1}{8}]$ in the above two $S_0$.

Theorems & Definitions (22)

• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Theorem 2.5
• proof