An iterative method for solving elliptic BVP in one-dimension

TL;DR

This paper introduces an improved iterative decomposition method for efficiently solving one-dimensional elliptic boundary value problems, demonstrating faster computation and guaranteed convergence under certain conditions.

Contribution

The paper proposes a novel iterative decomposition technique that enhances computational efficiency and provides theoretical convergence guarantees for solving elliptic BVPs in one dimension.

Findings

Method requires fewer computations for same accuracy.

Convergence is guaranteed for smooth forcing data.

Validated with problems having known solutions.

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.