Numerical Approximation of Laplace's Equation Using the Finite Element Method

Abstract

This article addresses the finding of an approximate solution to Laplace's equation, a basic elliptic partial differential equation, using the Finite Element Method (FEM). Laplace's equation, ∇²u = 0, plays a key role in the description of equilibrium of processes in physics and engineering, e.g., in steady state heat conduction, and in electrostatics. Whereas this type of solution is not always possible because of the restrictions it imposes on the geometry, FEM is a more general and flexible approach where complex geometries are to be treated We give the mathematical framework, weak format computation for FEM, the theory from existence and uniqueness to the regularity of its solution based on functional analysis (Lax–Milgram theorem). The FEM procedure (discretization employing basis functions and mesh generation) is explained with special focus on its influence on precision and efficiency. An important part of this work is the error and convergence analysis with computation of error estimates, confirming theoretical a priori error estimates (for instance, first order in (H¹ norm) and second order in (L² norm) for P1 elements) in numerical experiments. Visualization methods and error tables are employed to demonstrate features of solutions and measure accuracy. The tendency emphasizes the importance of mesh refinement techniques, in particular, adaptive methods based on a posteriori error estimation. This work validates the reliability and performance of the proposed FEM to obtain the solution of Laplace's equation, highlighting its significance in computational mathematics and engineering problems.