fix(poly): fixed numerous bugs related to inconsistent system sizing with the reduced operator

this has restored the symmetry which we relied on before.

src/poly/utils/public/utilities.h

@@ -3,9 +3,41 @@
 #include "mfem.hpp"
 
 namespace serif::utilities {
-    mfem::SparseMatrix buildReducedMatrix(
+    [[nodiscard]] mfem::SparseMatrix build_reduced_matrix(
         const mfem::SparseMatrix& matrix,
         const mfem::Array<int>& trialEssentialDofs,
         const mfem::Array<int>& testEssentialDofs
     );
+
+    /**
+     * @brief Generate a vector of 1s and 0s where 1 elemetns cooresponds to queried dofs. Useful for degugging
+     * @param allDofs array, counding from 0, of all dofs in the system
+     * @param highlightDofs the dofs that you want to identify
+     * @return
+     *
+     * *Example Usage:*
+     * One could use this to identify, for example, which dofs are being identified as the central dofs
+     * @code
+     * ...
+     * mfem::Array<int> phiDofs, thetaDofs;
+     * phiDofs.SetSize(m_fePhi->GetNDofs());
+     * thetaDofs.SetSize(m_feTheta->GetNDofs());
+     * const mfem::Vector phiHighlightVector = serif::utilities::build_dof_identification_vector(phiDofs, phiCenterDofs);
+     * const mfem::Vector thetaHighlightVector = serif::utilities::build_dof_identification_vector(thetaDofs, thetaCenterDofs);
+     * Probe::glVisView(
+     *     const_cast<mfem::Vector&>(phiHighlightVector),
+     *     *m_fePhi,
+     *     "Phi Center Dofs"
+     *     );
+     * Probe::glVisView(
+     *     const_cast<mfem::Vector&>(thetaHighlightVector),
+     *     *m_feTheta,
+     *     "Theta Center Dofs"
+     *     );
+     * @endcode
+     */
+    mfem::Vector build_dof_identification_vector(
+        const mfem::Array<int>& allDofs,
+        const::mfem::Array<int>& highlightDofs
+        );
 }