1D Finite Element Method (FEM) Solver

Description

This project implements a one-dimensional Finite Element Method (FEM) solver in Python. It utilizes the Galerkin method with 1D Lagrange basis functions and 2nd order Gaussian Quadrature for numerical integration. The solver is capable of handling problems using Forward Euler (FE) and Backward Euler (BE) methods for time-stepping.

This project was completed for COE 352 (Advanced Scientific Computing) at the University of Texas at Austin.

Features

• Solver Methods: Implements both Forward Euler and Backward Euler methods.
• Basis Functions: Uses 1D Lagrange basis functions for spatial discretization.
• Gaussian Quadrature: Employs 2nd order Gaussian Quadrature for numerical integration.
• User Interaction: Allows user input for nodes, timesteps, and solver methods.

Requirements

• Python 3.x
• NumPy
• Matplotlib

Usage

To run the solver, execute the fem1d.py script in a Python environment:

python3 fem1d.py

Follow the on-screen prompts to enter the number of nodes, the number of timesteps, and the preferred solver method (FE/BE).

Input Parameters

• N: Number of spatial nodes in the mesh.
• nt: Number of timesteps for the temporal discretization.
• method: Choice of solver method - 'FE' for Forward Euler or 'BE' for Backward Euler.

Output

The program computes and plots the numerical solution against the analytical solution at the specified timestep, providing a visual comparison.

Analysis

Note: There is sometimes an issue with the gifs where they won't replay. Please click on them and it will take you to another tab to view them.

We are focusing on teh the heat transfer problem

$$ u_t - u_{xx} = f(x, t), \quad (x, t) \in (0, 1) \times (0, 1) $$

with initial and Dirichlet boundary conditions

$$ u(x, 0) = \sin(\pi x), $$

$$ u(0, t) = u(1, t) = 0 $$

and function

$$ f(x, t) = (\pi^2 - 1)e^{-t}\sin(\pi x) $$

The analytic solution to this problem is

$$ u(x, t) = e^{-t}\sin(\pi x) $$

Question 2 - Forward Euler

Solve first by using a forward Euler time derivative discretization with a time-step of Δ𝑡 = 1/551 . Plot the results at the final time. Increase the time-step until you find the instability. What dt does this occur at? How does the solution change as N decreases?

The Forward Euler method has conditional temporal stability and must abide by the Courant number for convergence. The Courant number says that the time step size, Δt, must be sufficiently small relative to the square of the spatial grid size, Δx, for stability, especially significant in highly diffusive systems. Spatially, FE faces a truncation error, which decreases with finer spatial grids. The combination of spatial and temporal discretization affects the method's overall stability and accuracy, which requires delicate balance between small time steps for stability and finer grids for spatial accuracy.

Here are the final results, plotted at the final time, using the Forward Euler method with 551 time steps and 11 nodes. Image:

Here is a gif that shows many different time steps:

You might notice that the convergence of stability is very rapid, but happens at around a timestep of 1/280. We can zoom in and analyze it further. Slowed down and focusing on the interval of 265 - 285 time steps, we can see that the FE method gains stability around a time step size of dt = 1/278, or 278 time steps over our [0, 1] second interval:

gif:

One a fixed interval, the stability looks like this:

gif:

We can also see how the number of nodes, N, affects the solution. Here is a gif that shows how the plot looks as N increases from 2 to 11.

gif:

As you can see, the number of nodes greatly affects the accuracy of our numerical method, with the number of nodes inversly porportional to the error between the the analytical and numerical plots.

Question 3 - Backward Euler

Solve the same problem with the same time-steps using an implicit backward Euler. What happens as the time-step is equal to or greater than the spatial step size? Explain why.

The Backward Euler method offers unconditionally stable temporal characteristics, allowing for larger time steps without the risk of instability. This stability, however, comes at the cost of accuracy, as the method is only first-order accurate in time, leading to potential errors in scenarios with rapid temporal variations.

Here are the final results, plotted at the final time, using the Backward Euler method with 551 time steps and 11 nodes.

Image:

We can also see how the number of nodes, N, affects the solution. Here is a gif that shows how the plot looks as N increases from 2 to 11.

gif:

Image:

Lastly, here is a picture of a plot when the time-step is greater than the spatial step size

When the time-step Δt is large, specifically when it's comparable to or larger than the spatial step size, the solution's accuracy can degrade. This happens because the backward Euler method, while stable, is only first-order accurate in time, and large time-steps can lead to significant temporal discretization errors. As Δt increases, we can observe that that the solution becomes less responsive to rapid changes in the solution, which is a characteristic of the method's low temporal accuracy.