The variational approach of an elliptic problem and its solution by finite elements

TL;DR

This paper explores the variational formulation of an elliptic boundary value problem, establishing well-posedness using functional analysis, and demonstrates a numerical solution with finite element methods via freefem++.

Contribution

It introduces a variational approach for elliptic problems, analyzes their well-posedness, and provides a numerical solution implementation using freefem++.

Findings

The variational formulation ensures well-posedness of the elliptic problem.

The Lax-Milgram lemma is effectively used to analyze the problem.

Numerical solutions are successfully obtained with finite element methods.

Abstract

In this paper, we deal with an elliptic problem with the Dirichlet boundary condition. We operate in Sobolev spaces and the main analytic tool we use is the Lax-Milgram lemma. First, we present the variational approach of the problem which allows us to apply different functional analysis techniques. Then we study thoroughly the well-posedness of the problem. We conclude our work with a solution of the problem using numerical analysis techniques and the free software freefem++.