Poisson Operator¶
1D¶
\[\mathbf{\nabla}^2 f = q\]
\[\star \mathbf{d} \star \mathbf{d} f=q\]
Discrete form:
Primal:
\[\mathbf{\widetilde{H}}^{1}
(\mathbf{\widetilde{D}^{1}}
\mathbf{H}^{1}
\mathbf{D^{0}}
\bar{f}^{0} + a)=\bar{q}^{0}\]
where a is Neumann b.c.
Dual
\[\mathbf{H}^{1}
\mathbf{D^{0}}
\mathbf{\widetilde{H}}^{1}
(\mathbf{\widetilde{D}^{1}}
\tilde{f}^{0} + b)=\tilde{q}^{0}\]
where b is Dirichlet b.c.
Periodic grid 1D¶
\[\begin{split}\begin{eqnarray}
f(x) &=& e^{\sin x} \\
q(x) &=& e^{\sin x} (\cos^2 x - \sin x)
\end{eqnarray}\end{split}\]
(Source code, png, hires.png, pdf)
Chebyshev grid 1D¶
\[\begin{split}\begin{eqnarray}
f(x) &=& e^x \\
q(x) &=& e^x
\end{eqnarray}\end{split}\]
Boundary Conditions:
Dirichlet boundary conditions:
\[f(-1) = e^{-1} \quad f(+1) = e\]
Neumann boundary condition:
\[f^\prime(-1)= e^{-1} \quad f^\prime(+1)= e\]
(Source code, png, hires.png, pdf)
2D¶
\[(\star \mathbf{d} \star \mathbf{d} + \mathbf{d} \star \mathbf{d} \star ) f=q\]
Periodic grid 2D¶
\[f = e^{\sin(x)} \mathbf{d}x + e^{\sin(y)} \mathbf{d}y\]
Convergence for operator.
Chebyshev grid 2D¶
\[f = e^{x} \mathbf{d}x + e^{y} \mathbf{d}y\]
\[q = e^{x} \mathbf{d}x + e^{y} \mathbf{d}y\]
Convergence for operator.
