The Finite Element Method

Abstract

This chapter provides an overview of the finite element method (FEM) for the numerical solution of PDEs. It begins with FEM implementation for 1D systems, deriving the weak-form equivalent form of a time-dependent diffusion-type PDE example, then outlines the Galerkin method for its solution. System matrices are derived for 1D linear basis functions, comparing these with those obtained by the COMSOL FEM solver. The chapter then proceeds to describe higher-order 1D Lagrangian basis functions as well as cubic Hermite elements. Following these 1D methods, FEM is then described for 2D/3D PDEs, including higher-dimensional elements, as well as isoparametric elements for representing curved boundaries. The chapter concludes with FEM numerical implementation issues, as well as set of problems with fully-worked answers provided in the solution section.

Problems

5.1

Solve the following PDE numerically by-hand using FEM, utilising four equi-sized linear Lagrange elements over the interval $x \in [0,1]$, and compare the solution obtained with the exact solution.

$$\begin{aligned} \nabla \cdot \left( -\nabla u\right)= & {} 2 \nonumber \\ u(0)= & {} 1 \nonumber \\ u(1)= & {} -1 \nonumber \end{aligned}$$

You may use Matlab to solve the resulting $7\times 7$ system of equations.

5.2

Solve the following PDE numerically by-hand using FEM, utilising four equi-sized linear Lagrange elements over the interval $x \in [0,1]$, and compare the solution obtained with the exact solution.

$$\begin{aligned} \nabla \cdot \left( -\nabla u\right)= & {} x \nonumber \\ u(0)= & {} 1 \nonumber \\ \frac{\partial u}{\partial x}(1)= & {} -1 \nonumber \end{aligned}$$

You may use Matlab to solve the resulting $6\times 6$ system of equations.

5.3

For the family of cubic 1D Lagrange functions, we can define four cubic polynomial element shape functions $\varphi _i(\xi )$, $i=1\ldots 4$, such that

$$\begin{aligned} \varphi _i(\xi _j) = \left\{ \begin{array}{l@{\quad }l} 1 \quad &{} i=j \\ 0 &{} i \ne j \end{array} \right. \nonumber \end{aligned}$$

where $\xi _j$ is the local element coordinate of node j, and $i,j=1\ldots 4$. Assuming these nodes are equi-spaced along the element, determine the four cubic Lagrange functions and plot their shape.

5.4

Using 1D quadratic basis functions on a uniformly-spaced grid consisting of 4 elements, determine the finite element $\mathbf {D}$, $\mathbf {K}$ and $\mathbf {f}$ matrices for the 1D time-dependent diffusion equation

$$\begin{aligned} \frac{\partial c}{\partial t} = \frac{\partial ^2 c}{\partial x^2} \nonumber \end{aligned}$$

for $x \in [0,1]$ subject to zero-flux boundary conditions at $x=0$ and $x=1$, with initial value of c(x, t) at $t=0$ given by the square-wave distribution:

$$\begin{aligned} c(x,0) = \left\{ \begin{array}{ll} 1 \qquad &{} 0.4 \le x \le 0.6 \\ 0 &{} \mathrm {otherwise} \end{array} \right. \nonumber \end{aligned}$$

Using the same basis functions and number of elements, solve this PDE in COMSOL, plotting the solution at $t=0.1$, and confirm that the COMSOL-generated $\mathbf {D}$ and $\mathbf {K}$ matrices are equal to those you obtained analytically.