The Structure of the Capacitance matrix on domains with symmetry

Previously, we have investigated, by numerical and heuristic methods, the solutions defined by the eigenstates of the wavelet Capacitance Matrix[47] . The eigenstates are orthogonal on certain boundaries and induce some remarkable orthogonal fields that are solutions of the harmonic Helmholtz equation. The collection of solutions defined through symmetry is quite interesting and has several applications to the wavelet Capacitance Matrix method that are investigated in this report.

In this work we have developed a parallel domain decomposition method based on the eigenstates of the capacitance matrix for domains with symmetry. A complete description of the orthogonal subspace structures has been found, leading to efficient algorithms for the calculation of the eigenstates. The use of symmetry simplifies the matching conditions along the boundaries of elements. We define an adaptive algorithm by a natural expansion of basis elements by basis elements at a smaller scale. An extension of these results to three dimensions is found by a connection of symmetry with the crystallographic group of transformations.

The technique that we develop in this work applies to domains with the symmetry group of the regular n-gons whose vertices are determined by the nth rooth of unity, $z^n=1$. The differential operator is supposed to share this symmetry group, be translation invariant, and allow a reasonable definition of a Greens function .

Although our results do not depend of a specific operator or specific symmetry domain, we consider, for the sake of definiteness, the Helmholtz operator,

$$H_\alpha=-\Delta + \alpha$$

in a square domain, $\hat S$, with Dirichlet boundary conditions. There are common symmetries of operator and domain. These are rotation by $90^\circ$, reflection in diagonals and centerlines. The Wavelet-Galerkin discretization finds the Capacitance Matrix, $\hat C$. $\hat C$ is real, symmetric with complete, orthogonal set of $n$ eigenvectors on the discrete boundary, $\hat B_n$.

The set of eigenvectors, $\{\hat V_j,\;j=1,\cdots,n\}$, as boundary data, define solutions $\{\hat W_j,\;j=1,\cdots,n\}$ of the homogeneous, Helmholtz equation in the discrete periodic cell, $P$. The solutions $\hat W$ belong to five mutually orthogonal subspaces defined over the periodic cell. The five subspaces, $\{1,\;2,\;3,\;,4,\;5\}$, are related to different symmetries of the square. The subspaces are orthogonal over any domain in the periodic cell with the symmetry group of the square, $\hat S$.

To explicitly calculate the Green's Function we resolve the delta function in the space of translates of scaling function

$$\lambda_{x_0,y_0}(x,y)=\sum\sum\varphi(x_0-i)\varphi(x-i) \varphi(y_0-j)\varphi(y-j).$$

Since the translates of the scaling function are orthogonal and complete in $L^2$, the above expression implies that for a square integrable function $f$

$$f(x_0,y_0)=\int\int dxdy \lambda_{x_0,y_0}(x,y) f(x,y),$$

which is the definition of the delta function.

Therefore, we solve, by the wavelet-Galerkin method [47,48], the equation

$$\left(-\Delta + \alpha\right)G(x,x_0;y,y_0)=\lambda_{x_0,y_0}(x,y)$$

for the partial Green's Function, $G$. To find the usual Capacitance Matrix, $C$, we discretize the boundary into a series of points $\hat x_j$ and form the matrix whose $(i,j)$ component is $G(\hat x_i,\hat x_j)$. The evaluation of $G$ requires only one solution of the periodic, fast, wavelet-Galerkin solver [46].