Introduction

This is the companion repository to the paper, "Finite Element Exterior Calculus with Lower-order Terms", by Douglas N. Arnold and Lizao Li. It contains the complete numerical results and the source code for the numerical experiments.

Numerical results

The numerical results are collected in the directory named results. Raw output from the code conforms to the naming scheme

results-3d-[form degree]-form-deg[r].txt

where the form degree is either 1 or 2 and r is the polynomial degree for the FEEC family (please refer to the paper to see explicitly which polynomial degree is used for an individual variable given r for a particular element pair). Inside each file, every section looks like

[pair: ++   deg: 2   lots: FFFTF]
  h       sigma  rate        dsigma  rate  
---  ----------  ------  ----------  ------ 
  2  7.3338e-02          1.0180e+00          ....
  4  9.4712e-03  2.38    2.5864e-01  1.59  
  8  1.1064e-03  2.83    6.0866e-02  1.91  
 16  1.3089e-04  3.03    1.4532e-02  2.03  

In the header, pair ++ deg: 2 means that P3Lambda x P2Lambda elements are used. lots: FFFTF means that we only have a nonzero l4 lower-order term. The first column is the number of nodes per edge. For example, for the first row, we have a 2x2x2 random quasi-uniform mesh of the unit cube. The named columns are absolute L2 errors. For example, 7.3338e-02 is the L2 error in sigma. rate is computed from the successive rows.

The file rates_summary.txt gives a more readable summary of the raw output. Every section is of the form

[3d, 2-form, N2curl2 x RT2]
        3       2       2       2
l1      3       2       2       2
l2      2       2       2       2
l3      3       2       2       2
l4      3       2       2       2
l5      3       2       2       2

The header says that this is for the 2-form Hodge Laplacian on the 3D unit cube. The element pair used is Nedelec edge elements of the second kind of degree 2 and Raviart-Thomas elements of degree 2. The first column indicates which lower-order term is present (empty means no lower-order terms). The subsequent four columns are the convergence rates for sigma, dsigma, u, du, respectively. For example, the first row reads, without any lower-order terms, the L2-convergences rates are h^3, h^2, h^2, h^2, for sigma, dsigma, u, du, respectively.

Source code

To reproduce all the results in this repository and try other examples yourself, you can directly run the source code included in the directory named src.

First, you need a working FEniCS environment. It is an open source project and easy to install on Linux systems. Please refer to its official website http://fenicsproject.org/ for installation instruction. Alternatively, you can use the official Virtualbox image or Docker container image at https://github.com/FEniCS/docker.

In addition to that, you also need python packages tabulate and sympy. On most Linux systems, they can be installed by

sudo easy_install tabulate sympy

Both are open source and easy to install. tabulate is used to print pretty tables while sympy is used for the symbolic computation of the right-hand side and derivatives of the exact solution.

Once the environment is ready, the two scripts 3d-1-form.py, 3d-2-form.py can be executed by:

python 3d-1-form.py

or with MPI (recommended, these problems are pretty large):

mpirun -np 8 python 3d-1-form.py

Both scripts are well-documented. Basically, it will loop through all the cases specified at the end of the script and print out the numerical errors in tables mentioned in the [Numerical results] section. The lower-order terms are set in the make_data function.