Error Analysis of Heat Conduction Partial Differential Equations using Galerkin’s Finite Element Method

The present work explores an error analysis of Galerkin finite element method (GFEM) for computing steady heat conduction in order to show its convergence and accuracy. The steady state heat distribution in a planar region is modeled by two-dimensional Laplace partial differential equations. A simple three-node triangular finite element model is used and its derivation to form elemental stiffness matrix for unstructured and structured grid meshes is presented. The error analysis is performed by comparison with analytical solution where the difference with the analytical result is represented in the form of three vector norms. The error analysis for the present GFEM for structured grid mesh is tested on heat conduction problem of a rectangular domain with asymmetric and mixed natural-essential boundary conditions. The accuracy and convergence of the numerical solution is demonstrated by increasing the number of elements or decreasing the size of each element covering the domain. It is found that the numerical result converge to the exact solution with the convergence rates of almost O(h²) in the Euclidean L2 norm, O(h²) in the discrete perpetuity L∞norm and O(h1) in the H1 norm.