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 equation and system solvers, expanding the feasibility of complex simulations in physics, engineering, and other fields leveraging advanced hardware capabilities.

翻译：本文提出一种专为高效求解拟线性椭圆方程及方程组而优化的拟牛顿方法，特别侧重于基于GPU的计算。通过采用线性拉普拉斯算子与简化非线性项组合的方式近似雅可比矩阵，本方法在应对离散化偏微分方程所产生的大型稀疏矩阵时，降低了传统牛顿法典型的计算开销。我们还提供了收敛性分析，证明在正则化参数的最优选择下，方法能够局部收敛至精确解，从而确保每次迭代的稳定性与效率。二维及三维区域中的数值实验验证了所提方法结合张量积实现的鲁棒性与计算效益。该方法为加速拟线性椭圆方程及方程组的求解提供了可行路径，拓展了物理学、工程学及其他领域利用先进硬件能力进行复杂模拟的可行性。