MATHEMATICS

E. A. VOLKOV

EFFECTIVE ERROR ESTIMATES FOR SOLUTIONS BY THE GRID METHOD OF THE DIRICHLET PROBLEM FOR THE LAPLACE EQUATION ON POLYGONS

(Presented by Academician A. A. Dorodnitsyn, 19 XII 1963)

Known error estimates for the solution by the grid method of the Dirichlet problem for the Laplace equation, converging to the exact solution with rate \(h^k\), where \(h\) is the mesh step, \(1 \leq k \leq 6\), have been obtained, in particular, under the condition of boundedness of the \(k\)-th derivatives of the solution on the closed domain \((^{1-3})\). If the domain is a polygon \(M\) with an angle \(\beta\), \(\pi/k < \beta < \pi/(k-1)\), then, for arbitrarily prescribed boundary values that are as smooth as desired on the sides and continuous at the corner points, the \(k\)-th derivatives of the solution, as a rule, are unbounded on \(\overline M\) \((^4)\). This introduces a specificity into the construction of difference equations on a polygon and into the obtaining of error estimates. In the present note, estimates are first given for the derivatives of a harmonic function on a polygon and for the coefficients of the asymptotic representation of a harmonic function at the corners, expressed explicitly in terms of the maxima of the moduli of the derivatives of the function on the sides of the polygon. The estimates are obtained, relying on the results of \((^5)\), by methods of the theory of functions of a complex variable. Next, two types of error estimates are proposed for an approximate solution of the Dirichlet problem for the Laplace equation, obtained on a grid in the form of a solution of a system of difference equations composed of ordinary difference operators and special operators that take into account the asymptotics of the harmonic function at the corners. The first estimates are analytic, of Gershgorin-estimate type, but expressed in terms of known quantities. The other estimates, more accurate, are obtained in the form of a solution of an auxiliary system of difference equations. The method considered for solving the Dirichlet problem on a polygon, with sufficiently smooth boundary values, makes it possible to obtain an approximate solution on the grid converging to the exact solution with rate \(h^{m-2-\varepsilon}\), where \(4 \leq m \leq 8\).

1. Let a polygon \(M\) be given with boundary \(\Gamma\), having \(N\) sides. Let \(\Gamma_j,\ j=1,2,\ldots,\)—be the sides, including the corner points, numbered counterclockwise; \(\alpha_j\pi\) are the angles between \(\Gamma_j\) and \(\Gamma_{j+1}\), \(0<\alpha_j \leq 2\); \(P_j=\Gamma_j\cap\Gamma_{j+1}\). We shall say that \(f\in C_m(E)\) if \(f\) is \(m\) times continuously differentiable on \(E\). Consider the boundary-value problem

\[ \Delta u(x,y)=0 \text{ on } M,\qquad u=\varphi_j(s) \text{ on } \Gamma_j,\qquad j=1,2,\ldots,N, \tag{1} \]

where \(\varphi_j\in C_\nu(\Gamma_j)\), \(\nu \geq 0\). Denote
\(\Phi_j^k=\sup_{\Gamma_j}|\varphi_j^{(k)}|\);
\(\Phi=\max_j \Phi_j^0\);
\(\widetilde{\Phi}_j^k=\max\{\Phi_j^k,\Phi_{j+1}^k\}\);

\[ U^k(E)= \max_{\alpha+\beta=k}\sup_{0\leq\theta<\pi/2}\sup_E \left|\partial^{\alpha+\beta}u/\partial x_\theta^\alpha\,\partial y_\theta^\beta\right|^*, \]

where \(x_\theta,y_\theta\) are the variables in a coordinate system rotated through an angle \(\theta\); \(R_j\) is a rectangle with one side \(\gamma_j\subset\Gamma_j\) of length \(b\), and the second side of length \(a_j\), with the distances between the endpoints of \(\gamma_j\) and \(\Gamma_j\) not less than \(a_j\), and half of every circle of radius \(a_j\) with center on \(R_j\), lying above the diameter parallel to \(\Gamma_j\), in the inward direction—

* Below, analogous upper bounds for derivatives of functions denoted by the corresponding lowercase letters are denoted by other capital Latin letters.

of the normal to \(\Gamma_j\), belongs to \(\overline M\); \(S_j(r)\) is the sector of a circle of radius \(r\) with center \(P_j\), formed by the sides \(\Gamma_j\) and \(\Gamma_{j+1}\); \(T_j(r)\) is the arc of the sector \(S_j(r)\); \(M_\delta\) is the set of points \(M\) whose distance from \(\Gamma\) is greater than \(\delta\).

Lemma 1. The inequality \((2)\) holds
\[ U^\mu(\overline M_\delta)\leq \mu!4\Phi/\pi\delta^\mu,\qquad \mu\geq 0. \tag{2} \]

Lemma 2. If \(\varphi_j\in C_{\mu+1}(\Gamma_j)\), then
\[ U^\mu(\overline R_j)\leq \frac{2+2^{\mu+3/2}}{\pi(\mu+1)}\,a_j\Phi_j^{\mu+1} + 4\,\frac{\mu!}{\pi} \left( \frac{\Phi}{a_j^\mu} + \sum_{k=0}^{\mu}\frac{2^k\Phi_j^k}{k!\,a_j^{\mu-k}} \right) +\sqrt2\,\Phi_j^\mu . \tag{3} \]

Let \(P_j\) be the origin, and let the \(x\)-axis be directed along \(\Gamma_{j+1}\);
\(z=x+iy,\rho=|z|;\theta=\arg z;\ 1<\sigma\leq \nu;\ \varphi_t\in C_\nu(\Gamma_t);\ t=j,j+1;\)

\[ f_{j\sigma}= \begin{cases} c_{j0}\theta+\displaystyle\sum_{n=0}^{\sigma-1} d_{jn}\operatorname{Re} z^n +\displaystyle\sum_{n=1}^{\sigma-1} e_{jn}\operatorname{Im} z^n, & \alpha_j\ \text{irrational},\\[1.2em] \displaystyle\sum_{k=0}^{\sigma_*}{}' c_{jk}\operatorname{Im}\{z^{kq_j}\ln z\} +\displaystyle\sum_{n=0}^{\sigma-1} d_{jn}\operatorname{Re} z^n +\displaystyle\sum_{n=1}^{\sigma-1} e_{jn}\operatorname{Im} z^n, & \alpha_j=p_j/q_j, \end{cases} \]

where \(\sigma_*=[(\sigma-1)/q_j]\); \(p_j/q_j\) is an irreducible fraction; \(\sum'\) is a sum not extended to \(n\equiv0\pmod {q_j}\); \(c_{jn},d_{jn},e_{jn}\) are numbers determined by the conditions: \(\varphi_t-f_{j\sigma}|_{\Gamma_t}=o(\rho^{\sigma-1});\ t=j,j+1\).

Denote:
\[ \omega_j(k)= \max_{\sum_p k_p=k} \prod_p\left(\prod_{q=0}^{k_p-1}|\alpha_j-q|\right); \qquad \Lambda_j^\chi= 4\,\frac{\Phi+F_{j\nu_j}^{0}(T_j(r_j))} {\pi+r_j^{\chi/\alpha_j}(1-\lambda_j^{1/\alpha_j})^\chi} + \]
\[ +\widetilde\Phi_j^{\nu_j}r_j^{\nu_j-\chi/\alpha_j} \left( \frac{4}{\pi(1-\lambda_j^{1/\alpha_j})^\chi} \sum_{k=0}^{\chi}\frac{\omega_j(k)}{(\nu_j-k)!} + \frac{2+2^{\chi+3/2}}{(\nu_j-\chi-1)!\pi}\omega_j(\chi+1) + \frac{\sqrt2\,\omega_j(\chi)}{(\nu_j-\chi)!} \right), \]
\[ \Psi_j^\mu(\chi,\rho) = 4\rho^{\chi/\alpha_j-\mu}\mu!\, \frac{(1+\tau_j)^{\chi/\alpha_j}}{\pi\tau_j^\mu} \Lambda_j^\chi + \]
\[ + \rho^{\nu_j-\mu}\widetilde\Phi_j^{\nu_j} \left( \frac{(2+2^{\mu+3/2}\tau_j(1+\tau_j)^{\nu_j-\mu-1})} {(\nu_j-\mu-1)!(\mu+1)\pi} + \frac{\sqrt2}{(\nu_j-\mu)!} + \frac{\mu!}{\pi}\sum_{k=0}^{\mu} \frac{2^{k+2}}{(\nu_j-k)!\,k!\,\tau_j^{\mu-k}} \right). \]

Lemma 3. Let \(r_j>0;\ S_j(r_j)\subset \overline M;\ \sigma(k)=[(k+1)/\alpha_j];\ \chi_j\geq1;\ m\geq0;\ \nu_j=\max\{\sigma(\chi_j)+1,\chi_j+1,m+1\};\ \varphi_t\in C_{\nu_j}(\Gamma_t);\ t=j,j+1;\ 0<\lambda_j<1;\)
\(\tau_j=\min\{\sin(a_j/2),\,1/\lambda_j-1\}\) for \(\alpha_j<1/2;\)
\(\tau_j=\min\{\sqrt2/2,\,1/\lambda_j-1\}\) for \(\alpha_j\geq1/2;\)
\(r_j^*=r_j\lambda_j/(1+\tau_j)\); then on \(S_j(r_j^*)\)
\[ u=f_{j\nu_j}+g_j=f_{j\nu_j}+v_j+w_j; \tag{4} \]
\[ v_j=\sum_{k=1}^{\chi_j-1}\beta_{jk}\rho^{k/\alpha_j}\sin\frac{k\theta}{\alpha_j}, \quad(\chi_j>1); \qquad v_j=0\quad(\chi_j=1); \tag{5} \]
\[ |\beta_{jk}|\leq \frac{(2^k+2k(\alpha_j\sigma(k)-k))\,r_j^{\sigma(k)-k/\alpha_j}} {(\sigma(k))!\,k(\alpha_j\sigma(k)-k)\pi} \,\widetilde\Phi_j^{\sigma(k)} + 4\,\frac{\Phi+F_{j\sigma(k)}^{0}(T_j(r_j))} {\pi r_j^{k/\alpha_j}}; \tag{6} \]
\[ G_j^\mu(T_j(\rho))\leq \Psi_j^\mu(1,\rho),\qquad W_j^\mu(T_j(\rho))\leq \Psi_j^\mu(\chi_j,\rho),\qquad 0\leq\mu\leq m. \tag{7} \]

1. Let \(S_j(r_j)\subset \overline M;\ 0<\lambda_j<1;\ \varkappa_j\geqslant 1;\ m=4,8;\ \bar\nu_j=\max\{\nu_{j-1},\nu_j\}\) \((\nu_0=\nu_N);\ \varphi_i\in C_{\nu_j}(\Gamma_j),\ j=1,2,\ldots,N;\ \delta>0;\ M\subset \bigcup_{j=1}^{N}R_j\cup S_j(r_j^*/2)\cup M_\delta\).

Construct on \(\overline M\) a square grid with mesh size \(h\) \((^{3})\), for which one can choose numbers \(r_{j\tau}\), \(\tau=0,1,2\), such that
\(2\varkappa_j h\leqslant r_{j0}<r_{j1}-h<r_j^*-2h;\ r_j^*/2+\chi_l^h\leqslant r_{j2}\leqslant r_j^*-\chi_l^h;\ r_{j0}\leqslant r_{j2},\ j=1,2,\ldots,N;\ \chi_l^h=h\max\{\sqrt2,t_l\}\), where \(t_l\) is a quantity independent of \(h\), defined below, \(S_p(r_{p2})\cap S_q(r_{q2})=\varnothing\) for \(p\ne q\), and additional requirements following from what follows are satisfied. Denote by \(\Pi_j^h\) the set of grid nodes on \(S_j(r_{j0})\); by \(\Omega_j^h\), the set of nodes on \(S_j(r_{j1}+h)\setminus S_j(r_{j1}-h)\); by \(M_{m_j}^h\), the set of nodes on \(S_j(r_{j2})\setminus S_j(r_{j0})\) which, together with the minimal square \(\overline E_m\) containing the \(m\) nearest nodes, belong to \(\overline M\); by \(\Gamma_{m_j}^h\), the set of the remaining nodes on \(S_j(r_{j2})\setminus S_j(r_{j0})\); by \(M_m^h\), the set of nodes on \(M\setminus\bigcup_{j=1}^{N}S_j(r_{j2})\) which, together with \(\overline E_m\), belong to \(\overline M\); and by \(\Gamma_m^h\), the set of the remaining nodes on \(M\).

Consider the system of difference equations

\[ u_h=A_m u_h\quad \text{on } M_m^h;\qquad u_h=B_l u_h\quad \text{on } \Gamma_m^h; \]

\[ u_h=f_{j\nu_j}+A_m(u_h-f_{j\nu_j})\quad \text{on } M_{m_j}^h;\qquad u_h=f_{j\nu_j}+B_l(u_h-f_{j\nu_j})\quad \text{on } \Gamma_{m_j}^h; \tag{8} \]

\[ u_h=f_{j\nu_j}+D_{\varkappa_j}(u_h-f_{j\nu_j})\quad \text{on } \Pi_j^h,\qquad j=1,2,\ldots,N, \]

where \(m=4,8;\ l\leqslant m;\ A_m\) is the known averaging operator over \(m\) points \((^{3})\), with
\[ |u-A_4u|\leqslant h^4 U^4(\overline E_4)/24;\qquad |u-A_8u|\leqslant h^8 U^8(\overline E_8)/7!2; \]
\(B_l\) is an interpolation operator (for \(l\leqslant 5\), see \((^{6})\)) such that
\[ |u-B_lu|\leqslant h^l C_l U^l(\overline E_l'), \]
where \(\overline E_l'\) is the minimal convex closed set containing the interpolation points;
\[ C_l=\max_{\Gamma_h}\sum |d_l|\, t^l/l!h^l;\qquad \Gamma_h=\bigcup_{j=1}^{N}\Gamma_{m_j}^h\cup\Gamma_m^h; \]
\(t\) is the distance to an interpolation node; \(t\leqslant ht_l\); \(t_l\) is a constant independent of the choice of grid; \(d_l\) is an interpolation coefficient, with \(\sum'|d_l|\leqslant 1-\varepsilon_l;\ \varepsilon_l>0\); \(\sum\) is the sum over all interpolation points, and \(\sum'\) is the sum over interpolation points belonging to \(M\); \(D_{\varkappa_j}\) is an operator satisfying the condition

\[ D_{\varkappa_j}v_j(P)\equiv \sum_{\Omega_j^h} a_{PQ}v_j(Q)=v_j(P), \]

where \(v_j\) is a function of the form (5) with arbitrary coefficients \(\beta_{jk}\), \(P\in\Pi_j^h,\ Q\in\Omega_j^h\), and \(r_{j0}\) is such that

\[ \max_{P\in\Pi_j^h}\sum_{\Omega_j^h}|a_{PQ}|\leqslant 1-\delta_j,\qquad \delta_j>0. \]

For sufficiently small \(h\), the existence of the operators \(D_{\varkappa_j}\) is obvious. Indeed, the coefficients \(\beta_{jk}\) of the function \(v_j\) are determined as the solution of a system of equations with determinant nonzero and with free terms that are the values of \(v_j\) at the points dividing \(T_j(r_{j1})\) into \(\varkappa_j\) equal parts. The determinant remains bounded below in modulus by a positive quantity in a finite neighborhood of the indicated points, in which, for small \(h\), grid nodes will be found. In practice, the \(\beta_{jk}\) are expressed through the values of \(v_j\) at nodes nearest to the uniformly spaced points on an arc of radius of order \(h\varkappa_j/\alpha_j\). Then, using (5), one computes \(a_{PQ}\) and determines a value \(r_{j0}\) of order \(r_{j1}\) and a set \(\Pi_j^h\) on which

\[ \sum_{\Omega_j^h}|a_{PQ}|\leqslant 1-\delta_j. \]

Denote

\[ \xi_m=\max\left\{U^m(\overline M_\delta),\ \max_j U^m(\overline R_j),\ \max_l G_l^m\bigl(S_j(r_j^*)\setminus S_j(r_{j0}-\sqrt2\,h)\bigr)\right\}, \]

\[ \eta_l=\max\{U^l(\overline{M}_{\delta}),\ \max_j U^l(\overline{R}_j),\ \max_j G_j^l(S_j(r_j^*)\setminus S_j(r_{j0}-ht_l))\}, \]

\[ \vartheta=\max_j\bigl((r_{j1}+h)^{\varkappa_j/\alpha_j}(1-\delta_j)+r_{j0}^{\varkappa_j/\alpha_j}\bigr)\Lambda_j^{\varkappa_j}/\delta_j. \]

Lemma 4. The inequality holds

\[ |u_h-u|\leqslant \vartheta+h^l C_l\eta_l/\varepsilon_l+h^{m-2}C_m'\rho_0^2\xi_m/\varepsilon^*, \tag{9} \]

where \(\varepsilon^*=\min\{\varepsilon_l,\min_j\delta_j\}\); \(\rho_0\) is the radius of a circle containing \(M\); \(C_4'=1/4!\); \(C_8'=30/9!\).

Using (2), (3), (7), we estimate the right-hand side of inequality (9) in terms of known quantities. Moreover, if \(l=m-2\), \(0<ah^{\theta_j}<r_{j0}\), \(r_{j1}<bh^{\theta_j}\), \(j=1,2,\ldots,N\), where \(\theta_j=\alpha_j(m-2)/(\varkappa_j-1+\alpha_j m)\), then, according to (2), (3), (7), and (9), \(u_h-u=O(h^{\mu_0})\), where \(\mu_0=\min_j\varkappa_j(m-2)/(\varkappa_j-1+\alpha_j m)\).

Using the methods of § 6 \((^2)\) and (7), one can show that for
\[ \theta_j=\alpha_j(m-2)/(\varkappa_j-1+\alpha_j(m-2)) \]
\[ \mu_0=\min\{m-2,\min_j \varkappa_j(m-2)/(\varkappa_j-1+\alpha_j(m-2))\}. \]

1. A more accurate error estimate than (9) is a majorant obtained as the solution of a system of equations differing from (8) in that the boundary values are taken equal to zero, the signs are changed for the negative coefficients of the operators \(D_{x_j}\) and \(B_l\), and into each equation there is introduced a free term which is the smallest possible estimate of the residual \(u\) for the corresponding equation (8). The compilation of equations (8), as well as the choice of \(r_{j1}\) for which, on a fixed grid, the error estimate is minimal, can be automated on a computer. The selection of \(r_{j1}\) can be carried out independently, in view of the weak mutual influence of the errors at the angles (see, for example, \((^8)\)).

2. At angles \(\alpha_j=1/q_j\), which possess special properties (see \((^{9,4})\)), \(v_j\) is a polynomial estimated by inequality (6). At these angles, when \(\varkappa_j/\alpha_j\geqslant m^*\) (\(m^*=m\) for \(\alpha_j>1\); \(m^*=m-2\) for \(\alpha_j\leqslant1\)), one may put \(r_{j0}=0\) and construct difference equations by means of the operators \(A_m\) and \(B_l\). In \((^6)\), \(B_l\) were obtained for \(l\leqslant5\), but, using in addition to the values of the boundary function the values of its derivatives, it is possible to obtain \(B_6\) with nonnegative coefficients. Operators of the type \(D_{x_j}\) are used in practice in \((^{10,11})\), but without an estimate of the error of the solution. The results obtained carry over to the Dirichlet problem in a multiply connected domain bounded by a polygonal line and may be used for other grids, for example, a variable one.

The author expresses sincere gratitude to S. M. Nikol’skii for his attention to the present work.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
9 XII 1963

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