# An Efficient Quasi-Newton Method with Tensor Product Implementation for Solving Quasi-Linear Elliptic Equations and Systems

## TL;DR

This paper introduces a new quasi-Newton method optimized for GPU computing to efficiently solve quasi-linear elliptic equations and systems.

## Contribution

A novel quasi-Newton method with tensor product implementation is proposed, reducing computational overhead for solving large-scale quasi-linear elliptic problems.

## Key findings

- The method reduces computational overhead by approximating the Jacobian matrix with linear Laplacian and simplified nonlinear terms.
- Numerical experiments in 2D and 3D domains confirm the method's robustness and computational gains.
- Convergence analysis shows local convergence under optimal regularization parameter choices.

## Abstract

In this paper, we introduce a quasi-Newton method optimized for efficiently solving quasi-linear elliptic equations and systems, with a specific focus on GPU-based computation. By approximating the Jacobian matrix with a combination of linear Laplacian and simplified nonlinear terms, our method reduces the computational overhead typical of traditional Newton methods while handling the large, sparse matrices generated from discretized PDEs. We also provide a convergence analysis demonstrating local convergence to the exact solution under optimal choices for the regularization parameter, ensuring stability and efficiency in each iteration. Numerical experiments in two- and three-dimensional domains validate the proposed method’s robustness and computational gains with tensor product implementation. This approach offers a promising pathway for accelerating quasi-linear elliptic equations and systems solvers, expanding the feasibility of complex simulations in physics, engineering, and other fields leveraging advanced hardware capabilities.