A new hybrid block collocation method for solving elliptic PDEs

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Applied Mathematics

Background:

• Elliptic partial differential equations (PDEs) are fundamental in modeling various scientific and engineering phenomena.
• Existing numerical methods for solving these PDEs often face challenges with accuracy, stability, and efficiency.
• Developing advanced numerical techniques is crucial for accurate and efficient simulation of complex systems.

Purpose of the Study:

• To introduce and analyze a novel hybrid block collocation method (NHBCM) for solving two-dimensional elliptic PDEs.
• To theoretically establish the accuracy, stability, and convergence properties of the proposed NHBCM.
• To demonstrate the practical applicability and superior performance of the NHBCM through numerical experiments.

Main Methods:

• The study employs a hybrid block collocation approach combined with polynomial approximation.
• Theoretical analysis is conducted to determine the order of accuracy, stability, and convergence.
• The NHBCM is implemented and tested on various linear and nonlinear elliptic PDEs.

Main Results:

• The NHBCM achieves a high accuracy of fifth-order.
• Theoretical analysis confirms the method's robustness in terms of stability and convergence.
• Numerical results show that the NHBCM significantly outperforms other compared numerical methods in terms of efficiency.

Conclusions:

• The newly developed NHBCM is a highly accurate and efficient numerical method for solving two-dimensional elliptic PDEs.
• The method's fifth-order accuracy and robust convergence properties make it a valuable tool for scientific and engineering applications.
• NHBCM offers a superior alternative to existing numerical techniques for a wide range of elliptic PDE problems.