Numerical approximation of partial differential equations with MFEM library

TL;DR

This paper discusses the implementation of finite element methods for PDEs using the MFEM library, demonstrating solutions for Laplace and Navier-Stokes equations across different dimensions and complexities.

Contribution

It provides a detailed revision of finite element formulations and compares different finite element spaces using the MFEM library for PDE solutions.

Findings

Successful application of finite element methods to Laplace and Navier-Stokes equations

Comparison of Lagrange and Raviart-Thomas finite element spaces

Demonstration of MFEM library's capabilities in complex PDE simulations

Abstract

We revise the finite element formulation for Lagrange, Raviart- Thomas, and Taylor-Hood finite element spaces. We solve Laplace equation in first and second order formulation, and compare the solutions obtained with Lagrange and Raviart-Thomas finite element spaces by changing the order of the shape functions and the refinement level of the mesh. Finally, we solve Navier-Stokes equations in a two dimensional domain, where the solution is a steady state, and in a three dimensional domain, where the system presents a turbulent behaviour. All numerical experiments are computed using MFEM library, which is also studied.