A THEORETICAL OUTLOOK ON COLLOCATION METHODS AND PARTIAL DIFFERENTIAL EQUATIONS

Abstract [English]

The collocation method, leveraging wavelet-based techniques, provides a robust numerical framework for solving partial differential equations (PDEs) with improved accuracy, efficiency, and adaptability. By utilizing compactly supported Daubechies scaling functions, this method ensures key mathematical properties such as orthogonality, compact support, and vanishing moments, facilitating high-accuracy approximations with minimal computational overhead. Wavelet methods excel in handling both linear and nonlinear PDEs, supported by fast algorithms for multiscale analysis, efficient preconditioning techniques, and the capacity for adaptive refinement. Despite challenges in managing boundary conditions, nonlinear operators, and irregular domains, the collocation method offers viable solutions. Boundary conditions are imposed directly in the physical space, avoiding instability from improper basis extensions. Nonlinear terms are efficiently computed without transitioning between physical and coefficient spaces. Additionally, hierarchical multiresolution analysis supports adaptive refinement, concentrating computational efforts in regions of high complexity or singularities. Theoretical insights underline the method's stability and convergence. Stability is maintained through compact support, hierarchical representation, and diagonal preconditioning, while convergence benefits from the scaling functions’ approximation capabilities and multiresolution framework. Numerical experiments in one- and two-dimensional domains demonstrate the collocation method's efficacy in solving a wide range of PDEs, overcoming traditional limitations of wavelet-based approaches. This method integrates advanced mathematical principles with practical computational strategies, establishing itself as a powerful tool for modern numerical analysis.