**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: **

- 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.

- 1.1
Abstract Problems
- 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 \calC^{0}and \calC^{1}

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

- 2.1
Basic aspects of the finite element method
- 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_{k}v|_{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

- 3.1
Interpolation theory in Sobolev spaces