Étude mathématique et numérique de l’équation de poisson avec des conditions mixtes

Abstract

This memoir is dedicated to the study and numerical approximation of solutions to the Poisson problem, a fundamental model of elliptic partial differential equations. In a first step, we established the theoretical framework by recalling Green’s formulas and introducing Sobolev spaces, the natural functional framework of variational analysis. We then demonstrated, using the Lax-Milgram theorem, the existence and uniqueness of the weak solution of the problem. In a second step, we developed a numerical approximation using the finite element method on a conforming triangular mesh, studying two types of elements : P1 elements (affine polynomials) and P2 elements (quadratic polynomials). For each, we constructed the canonical basis of shape functions, established the discrete variational problem, and showed that it reduces to solving a linear system AX = b, where the stiffness matrix A is symmetric positive definite, hence invertible.