Summary: We study the behavior of solutions of the Helmholtz equation \((- \Delta_{\text{disc} , h}-E) u_h= f_h\) on a periodic lattice as the mesh size \(h\) tends to 0. Projecting to the eigenspace of a characteristic root \(\lambda_h(\xi)\) and using a gauge transformation associated with the Dirac point, we show that the gauge transformed solution \(u_h\) converges to that for the equation \((P( D_x)-E)v=g\) for a continuous model on \(\mathbb{R}^d\), where \(\lambda_h(\xi) \to P(\xi)\). For the case of the hexagonal and related lattices, in a suitable energy region, it converges to that for the Dirac equation. For the case of the square lattice, triangular lattice, hexagonal lattice (in another energy region) and subdivision of a square lattice, one can add a scalar potential, and the solution of the lattice Schrödinger equation \((- \Delta_{\text{disc} , h}+ V_{\text{disc} , h}-E) u_h= f_h\) converges to that of the continuum Schrödinger equation \((P( D_x)+V(x)-E)u=f\).