Advanced Numerical Analysis

Code: MA771 | L-T-P-C: 4-0-0-8

MA771 Advanced Numerical Analysis [4-0-0-8] Prerequisites: MA561 Fluid Dynamics

Iterative methods for linear systems : Classical iterative methods (Jacobi, Gauss-Seidel and successive overrelaxation (SOR) methods), Krylov subspace methods; GMRES, Conjugate-gradient, biconjugate-gradient (BiCG), BiCGStab methods, preconditioning techniques, parallel implementations.

Finite difference method : Explicit and implicit schemes, consistence, stability and convergence, Lax equivalence theorem, numerical solutions to elliptic, parabolic and hyperbolic partial differential equations.

 

Approximate method of solution : Galerkin method, properties of Galerkin approximations, Petrov-Galerkin method, generalized Galerkin method.

 

The finite element method(FEM) : FEM for second order problems, one and two dimensional problems, finite elements(elements with a triangular mesh and a rectangular mesh and three dimensional finite elements), fourth-order problems, Hermite families of elements, isoparametric elements, numerical integration.

Texts/References:

1. D.S. Watkins, Fundamentals of Matrix Computations, second Edition, Wiley-interscience, New York, 2002.
2. L.N. Trefethen and David Bau, Numerical Linear Algebra, SIAM, 1997.
3. Joe D. Hoffman, Numerical methods for Engineers and Scientist, McGrow-Hill, 1993.
4. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer, 1994.
5. K. Atkinson and W. Han, Theoretical Numerical Analysis : A Functional Analysis Frame-work, Springer-Verlag, New York, 2001.
6. P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
7. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994.
8. C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1987.