fix(poly): added stabilization term

least squares stabalization term seems to have stabilized polytrope and mostly resolve the overshoot mode (in more non linear cases (n > 2) the mode does reapear; however, it is much less pronounced

src/poly/utils/public/polytropeOperator.h

@@ -152,6 +152,7 @@ public:
         std::unique_ptr<mfem::MixedBilinearForm> M,
         std::unique_ptr<mfem::MixedBilinearForm> Q,
         std::unique_ptr<mfem::BilinearForm> D,
+        std::unique_ptr<mfem::BilinearForm> S,
         std::unique_ptr<mfem::NonlinearForm> f,
         const mfem::Array<int> &blockOffsets);
 
@@ -286,19 +287,33 @@ private:
     std::unique_ptr<mfem::MixedBilinearForm> m_M; ///< Bilinear form M, coupling θ and φ.
     std::unique_ptr<mfem::MixedBilinearForm> m_Q; ///< Bilinear form Q, coupling φ and θ.
     std::unique_ptr<mfem::BilinearForm> m_D;      ///< Bilinear form D, acting on φ.
+    std::unique_ptr<mfem::BilinearForm> m_S;
     std::unique_ptr<mfem::NonlinearForm> m_f;     ///< Nonlinear form f, acting on θ.
 
     // --- Full Matrix Representations (owned, derived from forms) ---
     std::unique_ptr<mfem::SparseMatrix> m_Mmat;   ///< Sparse matrix representation of M.
     std::unique_ptr<mfem::SparseMatrix> m_Qmat;   ///< Sparse matrix representation of Q.
     std::unique_ptr<mfem::SparseMatrix> m_Dmat;   ///< Sparse matrix representation of D.
+    std::unique_ptr<mfem::SparseMatrix> m_Smat;
 
     // --- Reduced Matrix Representations (owned, after eliminating essential DOFs) ---
     std::unique_ptr<mfem::SparseMatrix> m_MReduced; ///< Reduced M matrix (free DOFs only).
     std::unique_ptr<mfem::SparseMatrix> m_QReduced; ///< Reduced Q matrix (free DOFs only).
     std::unique_ptr<mfem::SparseMatrix> m_DReduced; ///< Reduced D matrix (free DOFs only).
+    std::unique_ptr<mfem::SparseMatrix> m_SReduced; ///< Reduced S matrix (free DOFs only, used for least squares stabilization).
     mutable std::unique_ptr<mfem::SparseMatrix> m_gradReduced; ///< Reduced gradient operator (G) for the nonlinear part f(θ).
 
+    // --- Scaled Reduced Matrix Representations (owned, after eliminating essential DOFs and scaling by stabilization coefficients) ---
+    std::unique_ptr<mfem::SparseMatrix> m_MScaledReduced; ///< Scaled M matrix (free DOFs only, scaled by stabilization coefficient).
+    std::unique_ptr<mfem::SparseMatrix> m_QScaledReduced; ///< Scaled Q matrix (free DOFs only, scaled by stabilization coefficient).
+    std::unique_ptr<mfem::SparseMatrix> m_DScaledReduced; ///< Scaled D matrix (free DOFs only, scaled by stabilization coefficient).
+    std::unique_ptr<mfem::SparseMatrix> m_ScaledSReduced; ///< Scaled S matrix (free DOFs only, scaled by stabilization coefficient).
+
+
+    // --- Stabilization Coefficient --- (Perhapses this should be a user parameter...)
+    static constexpr double m_stabilizationCoefficient = -1.0;       ///< Stabilization coefficient for the system, used to more tightly couple ∇θ and φ.
+    static constexpr double m_IncrementedStabilizationCoefficient = 1.0 + m_stabilizationCoefficient; ///< 1 + Stabilization coefficient for the system, used to more tightly couple ∇θ and φ.
+
     // --- State Vectors and DOF Management ---
     mutable mfem::Vector m_state;                 ///< Full state vector [θ, φ]^T, including essential DOFs.
     mfem::Array<int> m_freeDofs;                  ///< Array of indices for free (non-essential) DOFs.