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.

Periodic

Dirichlet

Neumann

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.

Periodic

Dirichlet

Neumann

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

\[\hat{e}_{i}\qquad\hat{\theta}^{i}\]

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

\[J_{i}^{\phantom{j}k}J_{\phantom{j}k}^{j}=\delta_{i}^{j}\]

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

\[\hat{\theta}^{i}\delta_{ij}\hat{\theta}^{j}=dx^{i}\,g_{ij}\,dx^{j}\]
\[g_{ij}=J_{\phantom{m}i}^{m}\,\delta_{mn}\,J_{\phantom{n}j}^{n}\]

Transforming one-forms

\[\alpha=\alpha_{i}\,dx^{i}=\hat{\alpha}_{i}\,\hat{\theta}^{i}\]
\[\alpha_{i}=\hat{\alpha}_{j}J_{\phantom{i}i}^{j}\]

Transforming two-forms

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