From Hilbert Spaces to Animated Elliptic PDEs: My GitHub Gist on Numerical Methods & Functional Analysis
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Finding analytical solutions to Partial Differential Equations (PDEs) is often difficult, making Numerical Methods crucial for approximating solutions and gaining insights. This section explores various computational techniques, starting with an overview of Numerical Methods. It then delves into Finite difference methods for elliptic problems, explaining forward, backward, and central difference quotients for domain discretization and derivative approximation.
The discussion moves to Finite element methods for elliptic problems, highlighting their flexibility with complex geometries using piecewise polynomial basis functions like the hat function, isoparametric elements, and Barycentric coordinates. The Galerkin method and Galerkin orthogonality are central to formulating these approximations.
The section also covers Parabolic problems and time evolution, introducing methods like Crank-Nicolson for stable time discretization, and the numerical treatment of the wave equation. Key concepts for evaluating numerical solutions, such as discretization error (local and global), the structure of resulting linear systems (mass and stiffness matrices), and efficient solution techniques like the conjugate gradient method are discussed. Advanced techniques like the wavelet method, quasi-uniform meshes, Ritz projection, and Runge-Kutta methods are also briefly mentioned. This exploration provides the tools to transform PDEs into computable solutions for analyzing complex physical and engineering systems.
