Solving Partial Differential Equations with the Finite Element Method using Firedrake
Stefany Arevalo, Radoslav Vuchkov & Noemi Petra
Partial Differential Equations (PDEs) are equations that contain unknown functions and their partial derivatives and are used to model a wide variety of phenomena such as sound, heat, diffusion, fluid dynamics, elasticity, etc. While some of these equations can be solved analytically, in most cases one needs to implement a numerical method such as the finite element method (FEM) to solve them numerically. Depending on the complexity of the problem (e.g., inputs, boundary conditions, geometry, dimension) implementing FEM may not be trivial. Firedrake is an automated system for the solution of PDEs using FEM. It uses sophisticated code generation that makes the numerical solution of PDEs via FEM easily accessible to domain scientists. Firedrake provides an environment that expedites the solution of PDEs. In this project we investigate how to install and run Firedrake. In particular we focus on solving the Poisson problem numerically with linear and quadratic finite element basis functions and report on convergence results. The results show that the sequence of solutions is converging as expected based on the theory.
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