Variants of the Finite Element Method for the Parabolic Heat Equation: Comparative Numerical Study

TL;DR

This paper compares various weighted residual finite element method variants for solving the parabolic heat equation, highlighting their suitability and stability across different conditions.

Contribution

It provides a comparative analysis of finite element method variants, identifying the most effective approaches for specific types of parabolic equations.

Findings

Collocation and Least-Squares variants are suitable for first order systems.

Galernik/Least-Squares method is more diffusive but offers stability.

Different methods perform variably depending on Péclet numbers.

Abstract

Different variants of the method of weighted residual finite element method are used to get a solution for the parabolic heat equation, which is considered to be the model equation for the steady state Navier-Stokes equations. Results show that the Collocation and the Least-Squares variants are more suitable for first order systems. Results also show that the Galerkin/Least-Squares method is more diffusive than other methods, and hence gives stable solutions for a wide range of P\'eclet numbers.