# 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

Coordinate bases for vectors and forms

Non-coordinate orthonormal bases for vectors and forms

Transformations to orthornormal basis

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

Relationship to the metric

Transforming one-forms

Transforming two-forms