Course information

Introductory Example: The Dirichlet problem. Weak form. Numerical solution using the Finite Element Method. Boundary Value Problems and Galerkin Method: General weak form. Lax-Milgram Theorem. Galerkin Method. Error estimation. Variational form. Rayleigh-Ritz-Galerkin Method. Generalized derivatives and Sobolev spaces. Green s types. Elliptic boundary value problems. Existence and uniqueness. Mixed boundary conditions. Applications. Finite Element Methods for Elliptic Boundary Value Problems: One-dimensional finite elements. Piecewise polynomial functions. Cubic Hermite functions and splines. Two-dimensional and three-dimensional finite elements. Element-wise polynomial functions. Tensor product functions. Error estimates. Applications: Fluid flow, Heat flow, Various electrical potentials, Loaded beam, Loaded plate. Finite Difference Methods: Sturm-Liouville and Dirichlet problems. Heat equation. Wave equation. Compatibility, stability and convergence.