Different Boundary Conditions under Spectral DEC

To validate the code and ensure that it does what is expected of it, we start off with a simple example of a linear wave travelling in one direction.

The checkerboard that the surface is shaded with corresponds to the actual cellular complex.




Having validated the code with a simple example, now we can consider a more complicated wave form. In the examples, below we will start off with a Gaussian bump in the middle of the domain and see how it spreads out under different boundary conditions.




Locally Orthonormal Frame

At each point we will also store a matrix \(J\) such that \(J^{T}J=g\) that will map to a locally orthonormal basis. The hodge star operator will be implemented in that basis as

\[\begin{split}\left(\begin{array}{cc} 0 & -1\\ 1 & \phantom{+}0 \end{array}\right)\end{split}\]

Coordinate bases for vectors and forms

\[\frac{\partial}{\partial x^{i}}\qquad dx^{i}\]

Non-coordinate orthonormal bases for vectors and forms


Transformations to orthornormal basis

\[\hat{e}_{i}=J_{i}^{\phantom{j}j}\frac{\partial}{\partial x^{j}}\quad\hat{\theta}^{i}=J_{\phantom{i}j}^{i}\,dx^{j}\]

where \(J_{i}^{\phantom{j}j}\) and \(J_{\phantom{i}j}^{i}\) are transposed inverses of each other


The above property is required because the pairing of one forms with vector fields remains invariant

\[\hat{e}_{i}\hat{\theta}^{i}=\frac{\partial}{\partial x^{i}}dx^{i}\]

Relationship to the metric


Transforming one-forms


Transforming two-forms

\[\beta=\beta_{ij}\,dx^{i}\wedge dx^{j}=\hat{\beta}_{ij}\,\hat{\theta}^{i}\wedge\hat{\theta}^{j}\]