A Quadratic Optimization Framework for Solving Incompressible Navier-Stokes Equations in Arbitrary Domain

A quadratic optimization based framework is proposed for solving the incompressible Navier-Stokes equations in arbitrary domain using finite difference methods and meshfree methods. The objective of the optimization is to minimize the residual error arising from the finite difference discretization of the PDE. To handle complex boundaries effectively, constraints derived from meshfree methods are incorporated into the optimization framework. This hybrid approach leverages the structured simplicity of finite difference schemes for discretization while exploiting the flexibility of meshfree methods for imposing boundary conditions and handling complex geometries. The versatility and robustness of this new methodology is demonstrated by applying it to a range of practical problems, highlighting its ability to achieve accurate solutions without the computational overhead and complexity associated with traditional mesh generation tools and specialized solvers. This approach seeks to bridge the gap between structured grid-based methods and meshfree techniques, offering a more adaptable and computationally efficient solution strategy for complex PDE problems.