Course Details:

Unit I (Boundary value problems in 1D) [5 Lectures] weak formulation, Lax–Milgram lemma and well–posedness, finite element formulation, well-posedness of the discrete solution, error estimates in energy norm, Cea’s lemma, L2–norm error estimates, Aubin-Nitsche duality argument, computational implementation.

Unit II (Poisson equation in 2D) [10 Lectures] weak and discrete formulation for Dirichlet, Neumann, and Robin boundary conditions, conforming finite element method, triangulation and refinement (red, green, blue), Barycentric coordinate functions, error estimates, best approximation property, local to global assembly, computational implementation.

Unit III (Approximation in Banach spaces) [10 Lectures] limitations of Lax–Milgram theorem, inf-sup conditions, Banach-Nečas-Babuška theorems, stability, conformity, consistency, approximability, error analysis, Strang’s lemmas, saddle point problems, Stoke’s equation, mixed finite elements, Fortin criterion, Taylor-Hood method.

Unit IV (Lower–order error estimates) [6 Lectures] Aubin–Nitsche theorem, Goal oriented error control, max–norm and weighted norm estimates, Lp–estimates for linear and nonlinear problems.

Unit V (General theory of Galerkin methods) [11 Lectures] Sobolev spaces, abstract variational problem, minimisation problem, Babuška–Brezzi theorems, elliptic regularity, Galerkin method, compactness and embedding theorems, convergence analysis, polynomial spaces and interpolation theorems, affine mappings, nonconforming finite element methods, extension to time-dependent problems with heat example as an example.

Text Books:

1. Theory and practice of finite elements, Alexandre Ern and Jean-Luc Guermond, Springer-Verlag, 2004.
2. The finite element method for elliptic problems, Philippe G. Ciarlet, SIAM (Classics in applied mathematics).
3. Topics in functional analysis and applications, S. Kesavan, New Age International.

Reference Books:

1. The mathematical theory of finite element methods, Susanne C. Brenner, L. Ridgway Scott, Springer.
2. Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation, Beatrice Riviere, SIAM (Frontiers in Applied Mathematics).
3. Numerical approximation of partial differential equations, Soren Bartles, Springer.
4. Finite elements: theory, fast solvers, and applications in solid mechanics, Dietrich Braess, Cambridge University Press.
5. The gradient discretisation method, Jerome Droniou, Robert Eymard, Thierry Gallouet, CindyGuichard, Raphaele Herbin, Springer Cham
6. Sobolev spaces, Robert Adams, John Fournier, Elsevier (ISBN - 9780120441433)