Duration: 24 weeks

Number of lectures per week: 3 (including tutorials)

Assessment: Examination and Project (some years)

End-of-year Examination: One 3-hour examination

Description:

1. Elliptic Boundary Value Problems

   1.1 Abstract Problems
   The symmetric case. Variational inequalities
   The nonsymmetric case. The Lax-Milgram lemma

   1.2 Examples of elliptic boundary value problems
   The Sobolev spaces H^m(W). Green's formulas
   First examples of second-order boundary value problems
   The elasticity problem
   Examples of fourth-order problems: The biharmonic problem, the plate problem.

2. Introduction to the Finite Element Method

   2.1 Basic aspects of the finite element method
   The Galerkin and Ritz methods
   The three basic aspects of the finite element method. Conforming finite element methods

   2.2 Examples of finite elements and finite element spaces
   Requirements for finite element spaces
   First examples of finite elements for second order problems : n- Simplicies of type (k), (3¢)
   Assembly in triangulations. The associated finite element spaces n-Rectangles of type (k). Rectangles of type (2'), (3'). Assembly in triangulations
   First examples of finite elements with derivatives as degrees for freedom: Hermite n- simplicies of type (3), (3'). Assembly in triangulations
   First examples of finite elements for fourth-order problems: the Argyris and Bell triangles, the Bogner-Fox-Schmit rectangle. Assembly in triangulations

   2.3 General properties of finite elements and finite element spaces
   Finite elements as triples (K,P,S). Basic definitions. The P-interpolation operator
   Affine families of finite elements
   Construction of finite element spaces X_h. Basic definitions. The X_h-interpolation operator
   Finite elements of class \calC⁰ and \calC¹
   Taking into account boundary conditions. The spaces X_0h and X_00h

   2.4 General consideration on convergence
   Convergence familily of discrete problems
   Céa's lemma. First consequences. Orders of convergence

3. Conforming Finite Element Methods for Second Order Problems

   3.1 Interpolation theory in Sobolev spaces
   The Sobolev spaces W^m,p(W). The quotient space W^k+1,p(W)/P_k(W)
   Error estimates for polynomial preserving operators
   Estimates of the interpolation errors |v-P_kv|_m.q.k for affine families of finite elements

   3.2 Application to second-order problems over polygonal domains
   Estimate of the error ||u-u_h||_l,W
   Sufficient conditions for lim_{h® 0}||u-u_h||_l,W = 0
   Estimate of the error |u-u_h|_0,W. The Aubin-Nitsche lemma