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Mathematics
V. I. Lebedev
On the Dirichlet and Neumann Problems on Triangular and Hexagonal Grids
(Presented by Academician S. L. Sobolev on 12 XII 1960)
It is known \((^2)\) that on a hexagonal grid the local approximation of the Laplace operator by differences is performed using 4 points, but the error of this approximation is \(O(h)\), while on a triangular grid, although the error of the approximation of the Laplace operator by differences is of order \(O(h^4)\), the approximation is performed using 7 points. In the present note it is shown that the error between the approximate and sufficiently smooth exact solution arising in the boundary-value problems under consideration when the Laplace operator is replaced by a difference operator defined on a hexagonal grid is of order \(h^2\); this will follow from the fact that the residual in the approximation of the Laplace operator by differences is represented in divergent form and has opposite sign at neighboring points, up to terms \(O(h^2)\). It is also shown that from systems of equations whose solutions approximate, on a triangular grid, the solutions of the problems under consideration, one can pass to another system (for the \(h\)-conjugate function defined on a hexagonal grid), each equation of which contains only 4 unknowns.
Let \(U\) be a harmonic function, and \(V\) its conjugate, defined inside a simply connected domain \(\Omega\) in the plane \((x_1,x_2)\). Construct in the plane \((x_1,x_2)\) a triangular grid \(D_h\) with mesh size \(h=h_1\). Denote by \(\Omega_h\) the set of centers of the triangles; it forms a hexagonal grid with mesh size \(h_2=h_1/\sqrt{3}\). Denote by \(B_h\) the set of midpoints of the segments forming the grids \(D_h,\Omega_h\). Consider a quadruple of points of the sets \(\Omega_h,D_h\) lying in \(\Omega\): let the points \(A,B\in\Omega_h\) be two neighboring points, the points \(C,D\in D_h\) be at distance \(h_1/\sqrt{3}\) from the points \(A,B\), and let \(\mathbf n,\mathbf s\) be unit vectors in the directions, respectively, \(\overline{BA}\), \(\overline{DC}\), forming a right-handed coordinate system \((n,s)\). Then the equalities
\[ \frac{\partial U}{\partial n}=\frac{\partial V}{\partial s},\qquad \frac{\partial U}{\partial s}=-\frac{\partial V}{\partial n} \tag{1} \]
will be replaced inside \(\Omega\) by the approximate ones
\[ h_2^{-1}(u_A-u_B)=h_1^{-1}(v_C-v_D), \]
\[ h_1^{-1}(u_C-u_D)=h_2^{-1}(v_B-v_A). \tag{2} \]
The errors \(\eta=U-u,\ \zeta=V-v\) thereby admitted will satisfy the system of equations
\[ h_2^{-1}(\eta_A-\eta_B)-h_1^{-1}(\zeta_C-\zeta_D)=\psi_1, \]
\[ h_1^{-1}(\eta_C-\eta_D)+h_2^{-1}(\zeta_A-\zeta_B)=\psi_2, \tag{3} \]
where, in the local coordinate system,
\[ \psi_1(0,0)=\frac{h_2^2}{6}\frac{\partial^3}{\partial n^3}\,[U(\theta_1h_2,\theta_2h_1)] =-\frac{h_2^2}{6}\frac{\partial^3}{\partial s^3}\,[V(\theta_1h_2,\theta_2h_1)], \]
\[ \psi_2(0,0)=\frac{h_2^2}{6}\frac{\partial^3}{\partial s^3}\,[U(\theta_3h_2,\theta_4h_1)], \tag{4} \]
\[ |\theta_i|\leqslant 1,\quad i=1,2,3,4. \]
It is enough to consider only two cases for system (2):
1) For a function \(u\), defined on \(\Omega_h\), we solve the difference Dirichlet problem; then for the \(h\)-conjugate function \(v\), defined on \(D_h\), we obtain the difference Neumann problem. For the domain \(\Omega\), from the set \(D_h\) we distinguish the interior nodes \(D_h^1\) and the boundary nodes \(\Gamma_h^1\). The function \(v\) is defined on
\[
D_h^2=D_h^1+\Gamma_h^1.
\]
From the set \(\Omega_h^2\)—the centers of triangles all of whose vertices belong to \(D_h^2\)—we distinguish \(S_h^1\), the set of centers of those triangles for which two vertices belong to \(\Gamma_h^1\); let
\[
\Omega_h^1=\Omega_h^2-S_h^1.
\]
The function \(u\) is defined on \(\Omega_h^2\).
2) For a function \(v\), defined on \(D_h^2\), we solve the difference Dirichlet problem; then for the \(h\)-conjugate function \(u\) we obtain the difference Neumann problem. We regard the function \(u\) as defined on the set \(\Omega_h^2+S_h^2\), where \(S_h^2\in\Omega_h\) and is at distance \(h_2\) from \(\Omega_h^2\).
All newly introduced sets are assumed to lie in \(\Omega\). The values of \(u\) on \(S_h^1\) determine on \(\Gamma_h^1\) the value of \(\Delta v/\Delta n\) for \(v\), and the values of \(\Delta u/\Delta n\) on \(S_h^2\) determine on \(\Gamma_h^1\) the values of \(v\) up to an additive constant (*). In addition, as is easy to see, the function \(u\) satisfies at interior points the difference Laplace equation for the hexagonal grid:
\[
\Delta_h' u=0,
\]
while the function \(v\) satisfies the difference Laplace equation on the triangular grid:
\[
\Delta_h'' v=0.
\]
Thus we find that to each of our boundary-value problems for the function \(v\) there corresponds a conjugate boundary-value problem for the function \(u\).
We now estimate the quantity \(\eta\). To this end we write the expression for \(\Delta_h'\eta\) at a point \(x\in\Omega_h^1\). Let \(\mathbf n_i,\ i=1,2,3,\)—be the unit vectors issuing from the point \(x\) in the directions of the three points of the set \(\Omega_h^2\) adjacent to \(x\); then, substituting indices \(i\) for the derivatives in expression (4), we obtain
\[ \Delta_h'\eta=\frac{4}{3}h_2^{-1}\sum_{i=1}^{3}\psi_1\left(\frac{1}{2}h_2\mathbf n_i\right)=\psi_3,\qquad \eta\big|_{S_h^1}=\eta_0, \]
let
\[ |\eta_0|\leqslant \varepsilon, \]
and if \(\eta=\eta_1+\eta_2\), where
\[ \Delta_h'\eta_1=\psi_3,\qquad \eta_1\big|_{S_h^1}=0; \]
\[ \Delta_h'\eta_2=0,\qquad \eta_2\big|_{S_h^1}=\eta_0, \tag{5} \]
then
\[ |\eta_2|\leqslant \varepsilon. \]
From equations (5) we obtain that
\[ h_2^2\sum_{\Omega_h^1}\eta_1\Delta_h\eta_1 = h_2^2\sum_{\Omega_h'}\eta_1\psi_3. \]
Summing by parts on both sides of this equality and denoting by \(\eta_{1n_i}\) the divided differences of the function \(\eta_1\) in the direction \(\mathbf n_i\), we obtain the estimate
\[ h_2^2\sum_{B_h}\sum_{i=1}^{3}(\eta_{1n_i})^2 \leqslant c_1h_2^2 \left[ h_2^2\sum_{B_h}\sum_{i=1}^{3} \left(\frac{\partial^3 U}{\partial n_i^3}\right)^2 \right]. \]
from which we obtain that the right-hand side of this inequality will be of order \(O(h_2^2)\), if the third derivatives of the function \(U\) are bounded or have singularities of the form \(\rho^{-\lambda_k}\) at a finite number of boundary points \(s_k\), where \(\rho_k\) is the distance to \(s_k\), and \(0 \leqslant \lambda_k < {}^1\!/_{2}\); hence, as in (4), we obtain that in every interior subdomain of the domain \(\Omega\)
\[ \eta = O(h_2^2+\varepsilon). \]
Let \(U \subset H(4,A,{}^1\!/_{2})\) (1), and let the functions \(\eta'\), \(\xi_1\) satisfy system (3); define the function \(\eta'\) on \(S_h^1\) as follows: let \(s_0 \in S_h^1\) and \(\eta'(s_0)=0\), and at the remaining points of \(S_h^1\) define \(\eta'\) in the direction of a left traversal of \(S_h^1\) so that \(\Delta \xi_1/\Delta n=0\) on \(\Gamma_h^1\). This can be done, using equations (3), at all points of \(\Gamma_h^1\) except one; at this point we additionally require that \(\xi_1=0\); note that at it \(\Delta \xi_1/\Delta n=O(h_1^2)\). Then in \(\Omega_h^2\), \(\eta_1-\eta'=O(h_2^2)\), and at any point \(x\in D_h^1\)
\[ \Delta_h''\xi_1=\psi_4, \tag{6} \]
where \(\psi_4=O(h^{5/2})\).
Let \(s_i,\ i=1,2,3\), be the unit vectors issuing from the point \(x\) in the directions of the segments \(e_{ik}\), \(k=1,2,\ldots,N_i,\ i=1,2,3\), forming the triangular grid of the set \(D_h^2\), and let the angles between \(s_1,s_2\) and \(s_2,s_3\), measured counterclockwise, be equal to \(\pi/3\); then the quantity \(\psi_4\), using the fact that
\[ \sum_{i=1}^{3}\frac{\partial^4 V}{\partial s_i^4}=0, \]
can be represented as
\[ \psi_4=\sum_{i=1}^{3}\psi_{5i}, \]
where
\[ \psi_{5i}= \frac{2}{3h_1} \left( \psi_1\left(x+\frac{1}{2}h_1s_i\right) -\psi_1\left(x-\frac{1}{2}h_1s_i\right) \right) +\frac{h_1^2}{27}\frac{\partial^4 V(x)}{\partial s_i^4}, \]
and therefore, from equation (6), we obtain
\[ h_1^2\sum_{B_h}\sum_{i=1}^{3}(\xi_{1s_i})^2 = -2h_1^2\sum_{i=1}^{3}\sum_{k=1}^{N_i}\sum_{e_{ik}}\xi_1\psi_{5i}. \tag{7} \]
If \(U\subset H(4,A,{}^1\!/_{2})\), then \((^5)\)
\[ \psi_{ik}(y_1,y_2)=h_1\sum_{y_1}^{y_2}\psi_{5i}=O(h_1^{5/2}), \]
where \(y_1,y_2\) are any two points lying on the segment \(e_{ik}\). Consequently, performing in the sum over \(e_{ik}\) in expression (7) summation by parts, using the inequality \(2ab\leqslant \varepsilon a^2+\varepsilon^{-1}b^2,\ \varepsilon>0\), and also estimating \(h_1\sum_{\Gamma_h^1}\xi_1^2\) by
\[ h_1^2\sum_{B_h}\sum_{i=1}^{3}(\xi_{1s_i})^2, \]
we obtain
\[ h_1^2\sum_{B_h}\sum_{i=1}^{3}(\xi_{1s_i})^2=O(h_1^5), \]
and, taking (3) into account, we obtain
\[ h_2\sum_l(\eta'_{1l})^2=O(h^4), \]
where \(l\) is any broken line forming a hexagonal mesh, and \(\eta'_l\) are the divided differences of the function \(\eta'\) along \(l\); hence it follows that \(\eta' = O(h_2^2)\) and
\[ \eta = O(h_2^2+\varepsilon) \]
throughout the whole domain \(\Omega\).
2) Let \(\left.\xi\right|_{\Gamma_h^1}=\varepsilon_1(h_2)\); then, carrying out arguments analogous to case 1), we obtain that in every interior subdomain of the domain \(\Omega\)
\[ \eta = O\bigl(\varepsilon_1(h_2)+h_2^2\bigr) \]
and in the whole domain \(\Omega\)
\[ \eta = O\left(\frac{\varepsilon_1(h_2)}{h_2}+h_2^2\right). \]
If one uses the properties of the Green’s functions of the difference problems considered, these estimates can be obtained under weaker assumptions concerning the smoothness of the function \(U\). Analogous error estimates arising in the application of hexagonal meshes can be obtained for elliptic equations with variable coefficients, and also for some nonstationary equations, for example for the heat-conduction equation.
Received
10 XI 1960
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