Add elliptic eigenvalue global prom example

[ckendrick]

Adds a simple elliptic eigenvalue global pROM example based on MFEM example 11p.

Note: MFEM needs to be compiled with MFEM_USE_LAPACK enabled to run the online phase. This adds a -l option to the compile.sh script to build MFEM with lapack enabled.

Sample runs:

./elliptic_eigenproblem_global_rom -offline -p 2 -id 0 -a 0.40
./elliptic_eigenproblem_global_rom -offline -p 2 -id 1 -a 0.45
./elliptic_eigenproblem_global_rom -offline -p 2 -id 2 -a 0.55
./elliptic_eigenproblem_global_rom -offline -p 2 -id 3 -a 0.60

./elliptic_eigenproblem_global_rom -merge -p 2 -ns 4

./elliptic_eigenproblem_global_rom -fom -p 2 -a 0.50 -visit

./elliptic_eigenproblem_global_rom -online -p 2 -a 0.50

Example FOM output:

Number of unknowns: 289
 Eigenvalue 0: 0.04533314
 Eigenvalue 1: 0.11332411
 Eigenvalue 2: 0.11486387
 Eigenvalue 3: 0.18192
 Eigenvalue 4: 0.22964377
Elapsed time for assembling FOM: 1.475750e-03 second
Elapsed time for solving FOM: 3.390041e-01 second

Example online output:

spatial basis dimension is 289 x 20
Eigenvalue 0: = 0.048430949
Eigenvalue 1: = 0.12021157
Eigenvalue 2: = 0.12147847
Eigenvalue 3: = 0.19456504
Eigenvalue 4: = 0.24285855
Relative error of ROM solution for eigenvalue 0 = 0.068334316
Relative error of ROM solution for eigenvalue 1 = 0.060776586
Relative error of ROM solution for eigenvalue 2 = 0.057586388
Relative error of ROM solution for eigenvalue 3 = 0.069508795
Relative error of ROM solution for eigenvalue 4 = 0.057544708
Relative l2 error of ROM eigenvector 0 = 0.070430601
Relative l2 error of ROM eigenvector 1 = 1.996413
Relative l2 error of ROM eigenvector 2 = 0.077548522
Relative l2 error of ROM eigenvector 3 = 0.10685373
Relative l2 error of ROM eigenvector 4 = 0.1494834
Elapsed time for assembling ROM: 5.230334e-03 second
Elapsed time for solving ROM: 7.794170e-04 second

[siuwuncheung]

I'm curious if there are any reasons to have different kinds of options, "On" vs "Off", "Yes" vs "No", true vs false. Can we unify them?

[siuwuncheung]

Inviting @dreamer2368 as a reviewer to evaluate the changes in the build process in this PR.

[dreamer2368]

The setup procedure looks good in general. If LAPACK is required for this example, can we install mfem with LAPACK dependency in our docker container?

[siuwuncheung]

LGTM except some minor things. Thanks @ckendrick!

For boarder communication, I am able to reproduce the eigenvalues, but the L2 error of eigenvectors appear different. The signs of the eigenvectors seem to be sort of random, which occasionally gives 200% error. One example that I got is

Relative l2 error of ROM eigenvector 0 = 2.0185057
Relative l2 error of ROM eigenvector 1 = 0.090773417
Relative l2 error of ROM eigenvector 2 = 0.076616931
Relative l2 error of ROM eigenvector 3 = 1.9746465
Relative l2 error of ROM eigenvector 4 = 1.9912245

I'll follow up with this example in PR#277 to enable the followings:

1. Use consistent signs of eigenfunctions
2. Add boundary condition switch
3. Clean up to retain only 4 problems, i.e. poisson, reaction-diffusion in heterogeneous media, Schrodinger equation with harmonic potential, Schrodinger equation with negative Gaussian potential well
4. Extend to 3D examples
5. Change some filenames

[siuwuncheung]

I had approved this PR after reviewing the changes and reproducing the results. We need one more approval to get this PR merged.

[dylan-copeland]

It looks like the condition being checked here is
if (std::abs(t(i) - mesh_center) > 0.5 * domain_limit)
The current version is equivalent, but less clear.