This course is an introduction to the theoretical and computational aspects of the finite element method for the solution of boundary value problems for partial differential equations. Emphasis will be on linear elliptic, self-adjoint, second-order problems, and some material will cover time dependent problems as well as nonlinear problems. Topics include: Sobolev spaces, variational formulation of boundary value problems, natural and essential boundary conditions, Lax-Milgram lemma, approximation theory, error estimates, element construction, continuous, discontinuous, and mixed finite element methods, and solution methods for the resulting finite element systems.
- FEM for two point boundary value problems
- Brief introduction to Sobolev Spaces
- FEM for elliptic equations
- Approximation theory for FEM
- FEM for parabolic equations
The course will not follow closely a specific textbook. Here are some recommended references.
The mathematical theory of finite element methods. S.C. Brenner and L.R. Scott, Springer-Verlag 1994.
Numerical analysis of partial differential equations. C. Hall and T. Porsching, 1990.
Understanding and implementing the finite element method. M. Gockenbach, SIAM 2006.
Finite elements: Theory, fast solvers, and applications in solid mechanics, 3rd ed. D.Braess, Cambridge 2007.
The finite element method for elliptic problems. P. Ciarlet, SIAM 2002.