Advanced Numerical Analysis I

This course covers basic numerical methods in the graduate level. The main themes are numerical solutions of PDEs and iterative methods for linear systems. For numerical solutions of PDEs, some basic theories of finite element methods for elliptic and parabolic equations are discussed. The topics include Sobolev spaces, embedding and trace theorems, variational formulation, Lax-Milgram theorem, finite element spaces and their interpolation theories, convergence in H1 and L2 norms. For iterative methods for linear systems, the topics include Richardson, Jacobi, Gauss-Seidel and SOR methods, Krylov subspace methods, conjugate gradient method, preconditioning techniques, convergence theories, GMRES method.