Variational Formulations for Linear Boundary-Value Problems

Abstract

n this thesis , we construct a variational formulation to a two- dimensional Laplace- Dirichlet problem, by transforming the continuous problem ( CP ) into an integral formulation known as a variational problem ( VP ) in Sobolev Spaces . We state some theorems and lemmas for the existence and uniqueness of the solution of the variational problem ( VP ) . We prove the existence and uniqueness of the solution of the variational problem ( VP ) . We also use the hypothesis of Lax- Milgram theorem and the Ce’a’s lemma to estimate the approximation error between the exact and the approximate solution to the ( VP ) . We state some description of an ordinary finite elements most commonly used in applications of engineering Sciences . Finally as a case , we construct a variational formulation for a one- dimensional Dirichlet and Neumann problems using Lagrange finite elements P1 .