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

Area of Science:

• Numerical Analysis
• Scientific Computing
• Computational Physics

Background:

• Quasi-linear elliptic equations are fundamental in modeling physical phenomena.
• Traditional Newton methods face computational challenges with large, sparse matrices from discretized partial differential equations (PDEs).
• GPU acceleration is crucial for handling the demands of modern scientific simulations.

Purpose of the Study:

• To introduce a novel quasi-Newton method for efficient GPU-based computation of quasi-linear elliptic equations.
• To reduce the computational overhead compared to standard Newton methods.
• To provide theoretical and numerical validation of the method's performance.

Main Methods:

• Developed a quasi-Newton method approximating the Jacobian matrix using linear Laplacian and simplified nonlinear terms.
• Implemented a tensor product approach for efficient matrix handling.
• Conducted convergence analysis to ensure local convergence and stability.
• Performed numerical experiments in 2D and 3D domains.

Main Results:

• The proposed method significantly reduces computational overhead for solving quasi-linear elliptic systems.
• Convergence analysis confirms local convergence to the exact solution with optimal regularization parameters.
• Numerical experiments demonstrate the method's robustness and computational efficiency on GPUs.
• The tensor product implementation further enhances performance.

Conclusions:

• The developed quasi-Newton method offers an efficient and robust solution for quasi-linear elliptic equations on GPUs.
• This approach accelerates complex simulations in physics and engineering.
• It paves the way for broader applications leveraging advanced hardware capabilities.