Solving Elliptic Partial Differential Equations Using the Finite Element Method

This article explores the process of solving elliptic partial differential equations using the finite element method. The finite element method (FEM) is a numerical technique for computing solutions to partial differential equations. The method involves discretizing the domain into smaller subregions called finite elements, then solving the equation locally for each element. These local solutions are subsequently assembled to obtain the global solution over the entire domain. The implementation typically follows these key steps: domain discretization using meshing algorithms, formulation of weak variational statements, construction of shape functions over each element, assembly of global stiffness matrices, and solution of the resulting linear system using numerical solvers. In this article, we focus on the practical implementation of FEM for elliptic equations, providing concrete examples that demonstrate the method's application. The discussion includes code-level details about handling boundary conditions, matrix assembly techniques, and efficient solving methods for large sparse systems commonly encountered in finite element analysis.