Finite Differences and Finite Elements: Variational Formulation

This lecture covers the concepts of finite differences and finite elements, focusing on their applications in numerical analysis. The instructor begins by discussing the finite difference method, explaining how it approximates solutions to differential equations. The lecture emphasizes the importance of formulating problems variationally, which allows for a more flexible approach to solving complex equations. The instructor introduces the Galerkin method, highlighting its historical context and relevance in engineering applications. The discussion includes the derivation of a linear system from the variational formulation, demonstrating how to transition from continuous to discrete problems. The lecture also addresses the significance of choosing appropriate basis functions for finite element methods, illustrating how these choices impact the accuracy and efficiency of numerical solutions. Throughout the session, the instructor engages with students, encouraging questions and clarifying complex concepts, ultimately providing a comprehensive overview of numerical methods in engineering and applied mathematics.