An optimal adaptive wavelet method for first order system least squares.

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Scientific Computing

Background:

• Second-order partial differential equations (PDEs) are fundamental in modeling various scientific phenomena.
• Solving complex PDEs often requires sophisticated numerical techniques.
• Existing methods may face challenges with efficiency or specific boundary conditions.

Purpose of the Study:

• To present a novel reformulation of any well-posed second-order PDE into a first-order least squares system.
• To introduce an adaptive wavelet solver for this reformulated system.
• To analyze the computational complexity and applicability of the proposed method.

Main Methods:

• Reformulation of second-order PDEs into first-order least squares systems.
• Development and application of an adaptive wavelet solver.
• Analysis of computational complexity and convergence properties.

Main Results:

• Demonstration that any well-posed second-order PDE can be converted into a well-posed first-order least squares system.
• The adaptive wavelet solver achieves optimal computational complexity.
• Successful application to second-order elliptic PDEs with inhomogeneous boundary conditions and stationary Navier-Stokes equations.

Conclusions:

• The proposed reformulation and adaptive wavelet solver offer an efficient and general approach for solving second-order PDEs.
• This method provides a unified framework for a range of challenging PDE problems.
• The findings have significant implications for computational science and engineering.