# The chain collocation method: A spectrally accurate calculus of forms¶

The figures below were used in the paper published here.

## Figure 3¶

Examples of periodic interpolators for $$N=6$$ (a) and $$7$$ (b), and corresponding periodic histopolators (c) and (d), scaled by $$h=2\pi/N$$ for clarity. While the interpolator $$\alpha_N$$ satisfies $$\alpha_N(nh \mod 2\pi) = \delta_{0n} \; \forall n$$, the histopolator $$\beta_N$$ integrates to $$1$$ over the dual cell straddling $$x=0$$, and to $$0$$ over other dual cells in the range $$[0,2\pi]$$. Note that the alternating red and green colors are used to mark out dual cells, and to illustrate that the integral of $$\beta_N$$ over each of these dual cells sums to zero or one.

## Figure 5¶

Chebyshev primal basis functions for a grid with $$N=7$$. We normalize the one-form basis functions by $$x_n\!-\!x_{n-1}$$ to have approximately the same scale in our visualizations.

## Figure 6¶

Chebyshev dual basis functions for a grid with $$N=7$$.

## Figure 7¶

Convergence graphs for a 1D Poisson equation: (a) we solve $$\Delta f = e^{\sin x} (\cos^2 x - \sin x)$$ on a periodic domain for either a primal, or dual $$0$$-form $$f$$; (b) we solve math:Delta f = e^{x} on a Chebyshev grid, for either a primal $$0$$-form with Dirichlet boundary conditions $$f(-1)=e^{-1}$$ and $$f(1)=e$$, or for a dual $$0$$-form with Neumann boundary conditions math:f^prime(-1)= e^{-1} and $$f^\prime(+1)= e$$. All of our results exhibit spectral convergence (measured through the $$L^\infty$$ error $$\|f - \Delta^{-1}q \|_\infty$$), with the conventional plateau when we reach the limit of accuracy of the representation of floating point numbers.

## Figure 8¶

Convergence graphs for a 2D Poisson equation; (a) we solve $$\Delta f = e^{\sin x} (\cos^2 x - \sin x)+e^{\sin y} (\cos^2 y - \sin y)$$ on a periodic domain for either a primal, or dual $$0$$-form $$f$$; (b) Now for $$\Delta f = e^{\sin x} (\cos^2 x - \sin x) \mathbf{d}x +e^{\sin y} (\cos^2 y - \sin y) \mathbf{d}y$$; (c) we solve $$\Delta f = e^{x}+e^{y}$$ on a Chebyshev grid, for either a primal $$0$$-form with Dirichlet boundary conditions $$f(x,y)=e^{x}+e^{y}$$, or for a dual $$0$$-form with Neumann boundary conditions $$\nabla f(x) \cdot \mathbf{n} = (e^x\;e^y)^t \;\mathbf{n}$$; (d) Now for $$\Delta f = e^{x} \mathbf{d}x+e^{y}\mathbf{d}y$$. All of our results exhibit spectral convergence (measured through the $$L^\infty$$ error $$\|\Delta f - q \|_\infty$$), with the conventional plateau in accuracy for fine meshes.

## Figure 9¶

Plot for the solution of $$(\star\mathbf{d}\star\mathbf{d}+\mathbf{d}\star\mathbf{d}\star)f\!=\!0$$ with the boundary conditions of $$\star f |_{\mathcal{B}} = 0$$ (vector field is tangent to the boundary) and $$\star\mathbf{d}\star f |_{\mathcal{B}}\!=\!1$$ (the curl at the boundary is equal to $$1$$). The domain is $$[-1,1]^2$$, discretized by a $$10\!\times\!10$$ Chebyshev grid.