# The Discretized Laplace Operator on Hyperrectangles with Zero Dirichlet Boundary Conditions

In this blog post, I present stiffness and mass matrix as well as eigenvalues and eigenvectors of the Laplace operator (Laplacian) on domains $(0, \ell)$, $(0, \ell_1) \times (0, \ell_2)$, and so on (hyperrectangles) with zero Dirichlet boundary conditions discretized with the finite difference method (FDM) and the finite element method (FEM) on equidistant grids. For the FDM discretization, we use the central differences scheme with the standard five-point stencil in 2D. For the FEM, the ansatz functions are the hat functions. The matrices, standard eigenvalue problems $A v = \sigma v$, and generalized eigenvalue problems $K w = \tau M w$ arising from the discretization lend themselves for test problems in numerical linear algebra because they are well-conditioned, not diagonal, and the matrix dimension can be increased arbitrarily.

Python code generating the matrices and their eigenpairs can be found in my git repository discrete-laplacian.

# CMake: "Errors occurred during the last pass"

You executed

ccmake -G ninja /path/to/source-code

on the command line and after pressing

[c] to configure

ccmake presents you an empty window with the message

Errors occurred during the last pass

in the status bar. The cause of this problem is the misspelled generator name on the ccmake command line (argument -G): the generator is called Ninja with capital N as the first letter.