An iterative method for solving elliptic BVP in one-dimension

TL;DR

This work addresses solving a 1D elliptic BVP with variable conductivity $\kappa(x)$ by introducing a decomposition-based iterative method. Building on the Lu–Zhang approach, the authors reformulate the expansion so that higher-order terms depend only on $u_0$, resulting in a variational system that requires only two Poisson solves (in contrast to $M+1$ in the original method). They establish $H^1$-convergence of the approximation $u_M$ to the true solution as $M\to\infty$, with explicit bounds on the truncation error that depend on $\| abla\psi\|_{\infty}$ and $|u_0|_{H^1}$. Numerical experiments on several test problems demonstrate that the method achieves high accuracy with modest computational effort, validating both the theoretical convergence and practical efficiency, particularly for smooth coefficients.

Abstract

This paper presents a decomposition method for solving elliptic boundary value problems in one-dimension. The method is an improvement to an existing technique for approximating elliptic systems. It is demonstrated to be computationally superior to the original formulation as less computations are required to obtain an approximation of the same accuracy. Convergence of the method is justified and supported by some theoretical results. We show that for a sufficiently smooth forcing data, the method always converge for a relatively small truncation order. The method is tested using some problems with exact solutions.

Theorems & Definitions (14)

• Lemma 5.1
• proof
• Lemma 5.2
• proof
• Lemma 5.3
• proof
• Lemma 5.4
• proof
• Lemma 5.5
• proof