Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques

Figures & data

Figure 1. The solution domain Ω with boundary Γ = Γ_U ∪ Γ_C given by (37) and (38). The closure of the arcs Γ_U and Γ_C meet at the two points γ₁ and γ₂. A mesh of 40 quadrature panels is constructed on Γ, 10 of which are located on Γ_U. A source S₁, for the generation of Cauchy data via (39), is marked by ‘*’.

Figure 2. The double-step recursions (31) and (20) are run for 2k = 0, 2, 4, …, 10⁶ with the initial guesses f₀ = 0, f₀ = f_ref, g₀ = 0 and g₀ = g_ref. The symbols ‘○’ and ‘*’ refer to f_k, while ‘⋄’ and ‘+’ refer to g_k.

Figure 3. Reconstruction of the Dirichlet data f_k and Neumann data g_k with the double-step recursions (31) and (20) at 2k = 93,260. The initial guesses are f₀ = 0 and g₀ = 0, respectively.

Figure 4. The double-step recursion (31) is run for 2k = 0, 2, 4, …, 10⁶. The initial guess f₀ is either zero or equal to a discretized matching polynomial p^(m) of degree 2m + 1.

Figure 5. Reconstruction of Dirichlet data f_k and Neumann data g_k+1/2 with the double-step recursion (31) at 2k = 93,260. The initial guess is f₀ = p⁽³⁾. A single recursion step (33) is taken to get from f_k to g_k+1/2.

Figure 6. The projected double-step recursion (50) is run with n = 15 for 2k = 0, 2, 4, …, 10⁸. The initial guess is either zero or equal to a transformed discretized matching polynomial of degree 2m + 1.

Figure 7. The direct method for f and g via (54), (55) and g = B_NDf + G_NCg_C.

Figure 8. Reconstruction of the Dirichlet data f and Neumann data g via (54), (55) and g = B_NDf + G_NCg_C at n = 15.