Convergence rate for numerical computation of the lattice Green’s function

Flexible boundary-condition methods couple an isolated defect to bulk through the bulk lattice Green’s function. Direct computation of the lattice Green’s function requires projecting out the singular subspace of uniform displacements and forces for the infinite lattice. We calculate the convergence rates for elastically isotropic and anisotropic cases for three different techniques: relative displacement, elastic Green’s function correction, and discontinuity correction. The discontinuity correction has the most rapid convergence for the general case.

Figure 1

(Color online) Integrand of the inverse Fourier transform for (a) LGF, (b) relative displacement method, (c) EGF correction, and (d) discontinuity correction at $\vec{R}=\left(1,1\right)$. ${\underset{̰}{G}}^L\left(\vec{k}\right)$ has a second-order pole at the $\Gamma$ point. The relative displacement method avoids the pole by considering only the displacements relative to a fixed point. The EGF correction removes the second-order pole by subtracting a cutoff elastic Green’s function. Removal of the pole creates a discontinuity independent of $\left|\vec{k}\right|$ at the $\Gamma$ point. The discontinuity correction removes the discontinuity created by EGF correction. The remaining part of the integrand is smooth in the entire Brillouin zone. The bottom row shows the variation in the integrand as a function of $\left|\vec{k}\right|$ when the origin is approached from different angles $\theta ={\text{tan}}^{-1}\left(k_y/k_x\right)$. The discontinuity created at the $\Gamma$ point by the relative displacement method and EGF correction is independent of $\left|\vec{k}\right|$ but depends on the direction of approaching the origin. N.B.: the vertical scale changes from LGF to relative displacement and EGF correction to discontinuity correction.

Figure 2

(Color online) Convergence rate with number of $k$ points of the relative displacement method, EGF correction, and discontinuity correction in a 2D square lattice. We expect $N_k^{-2}$ convergence for discontinuity correction, and poorer $N_k^{-1}$ convergence for EGF correction and relative displacement method. Using a non-$\Gamma$ centered mesh (left) causes an unusually fast convergence for the relative displacement method in elastically isotropic materials. The exponents in the power-law scalings are not affected by the value of $\vec{R}$ while prefactors are changed in relative displacement method and are of the same order in EGF and discontinuity corrections.

Figure 3

(Color online) Convergence rate with number of $k$ points of the relative displacement method, elastic GF correction, and discontinuity correction in a 3D cubic lattice. The error for discontinuity correction method scales as $N_k^{-4/d}$ with dimension $d$ equal to three. Note that using a non-$\Gamma$ centered mesh creates a faster convergence for the relative displacement method as observed in the 2D case.

Figure 4

(Color online) Convergence rate with number of $k$ points of the relative displacement method, elastic Green’s function correction, and discontinuity correction in computation of the $G_{11}$ component of a 2D LGF in Al. The convergence of LGF calculations in a fcc lattice is the same as the one observed in the simplified problem in agreement with the expected values. Note that use of the non-$\Gamma$ centered mesh does not cause a fast convergence for relative displacement method due to the anisotropy of the elastic Green’s function.