On the Truncation Error of Discrete Approximations to the Solutions of Dirichlet Problems in a Domain with Corners

The solution for a Dirichlet problem on a given plane domain and with given boundary values is usually approximated in numerical computation by its discrete analog defined and determined on an approximating set of net points. It can be proved that the approximation thus obtained converges to the exact solution, when the net becomes denser indefinitely, independently of the domain and the boundary values subject to rather weak conditions. Nevertheless, the irregularity of the boundary curve and of the boundary values affects strongly the convergence rate. For instance, if both are analytic and if a proper boundary interpolation scheme is used, then the simplest net analog leads to an error which decreases asymptotically at least proportional to the square of the mesh constant <italic>h</italic>, as proved by Gerschgorin [2] and Collatz [1]: δ<subscrpt><italic>h</italic></subscrpt> = <italic>O</italic>(<italic>h<supscrpt>2</supscrpt></italic>). In introducing the convergence exponent (1) <italic>&kgr;</italic> = lim<subscrpt><italic>h</italic>→0</subscrpt> inf max<subscrpt><italic>P<subscrpt>h</subscrpt>CD<subscrpt>h</subscrpt></italic></subscrpt> log | δ<subscrpt><italic>h</italic></subscrpt>(<italic>P<subscrpt>h</subscrpt></italic>) |/log <italic>h</italic>, <italic>D<subscrpt>h</subscrpt></italic> being the approximating net domain with the mesh constant <italic>h</italic> and <italic>P<subscrpt>h</subscrpt></italic> its node, this result can be expressed also by the inequality <italic>&kgr;</italic> ≧ 2. On the other hand, the author of this paper has shown in [4] that if the boundary of the domain is piecewise analytic, i.e., composed of a finite number of analytic arcs, and the boundary values are analytic on each closed arc, but jump discontinuities at the join of two arcs are permitted, then the convergence exponent depends on the angles at those corners where two adjacent arcs are connected. (Of course, the analytic character of the prescribed boundary values may alter at a finite number of interior points of an analytic boundary arc; it is sufficient to partition this arc at these points into different arcs with angles π at the related corners.) Actually, if απ is the largest of these angles, then, by using, for instance, the extrapolation scheme proposed by Collatz in [1] in order to determine the boundary values for the approximating net domain, one obtains the inequality (2) <italic>&kgr;</italic> ≧ min (2, 1/α). In the paper [4] it has been proved, moreover, that if the boundary values are not only piecewise analytic but also continuous, then for α < 1 the relation δ<subscrpt><italic>h</italic></subscrpt> = <italic>O</italic> (<italic>h<supscrpt>2</supscrpt></italic>) holds. This implies that (3) <italic>&kgr;</italic> ≧ 2, (α < 1). However, no information is given in the case where α ≧ 1, i.e., if either the domain has corners where two different analytic arcs are connected to form an angle greater than or equal to π, or at some interior point of an analytic boundary are the analytic character of the prescribed boundary values changes. The result (2) is valid, of course; however, since (3) is an essentially better result than (2) for 1/2 < α < 1, one could expect that some improvement is possible also in the cases α ≧ 1, if the prescribed values are continuos. The results obtained theoretically in the paper [4] may be summarized as follows. If the domain has a piecewise analytic boundary and the boundary values are also piecewise analytic, both in the sense described above, then (2) <italic>&kgr;</italic> ≧ min (2, 1/α), (α > 0), where the boundary values may be discontinuous. However, if the boundary values are continuous, then (3) <italic>&kgr;</italic> ≧ 2, (0 < α < 1). The purpose of the present paper is to develop further the considerations in [4] and to demonstrate those theoretical results by some experimental ones. These seem to indicate that in the two inequalities above, at least in some cases, the equality sign holds, and, moreover, that the latter theoretical rule is just a part of a more general hypothetical rule, namely (4) <italic>&kgr;</italic> ≧ min (2, 2/α), (α > 0). Table I presents the results of some experiments with discontinuous but piecewise analytic boundary values. In each case the domain <italic>D</italic> is a polygon such that its boundary <italic>B</italic> contains the boundary nodes set of the approximating square net domain <italic>D<subscrpt>h</subscrpt></italic>. Moreover, the boundary values are assumed to vanish, except at those corner points with the greatest angle απ, and, hence, the asymptotic solution for <italic>h</italic> → 0 is known to be identically zero. It is true that the boundary values thus defined do not fall strictly under the definition of piecewise analytic boundary functions, given above, but they can be interpreted as the difference of two such admissible functions. If <italic>u<subscrpt>h</subscrpt></italic> and <italic>u<subscrpt>h</subscrpt></italic> are the discrete solutions of the related problems, then it is obvious that <italic>u<subscrpt>h</subscrpt></italic> is their difference, and its rate of convergence, as <italic>h</italic> → 0, is at least the smaller of those for <italic>u<subscrpt>h</subscrpt></italic> and <italic>u<subscrpt>h</subscrpt></italic>. Accordingly, the rate of convergence obtained by using such degenerate boundary functions is now an upper bound for the rate originally investigated. Now, if the truncation error were exactly δ<subscrpt><italic>h</italic></subscrpt> = <italic>u<subscrpt>h</subscrpt></italic> - <italic>u</italic> = <italic>Ch<supscrpt>&kgr;</supscrpt></italic>, with some <italic>C</italic> independent of <italic>h</italic>, then for two different values <italic>h</italic> with the ratio 2 (for instance <italic>h</italic> = 2 and <italic>h</italic> = 1) the corresponding approximations <italic>u</italic><subscrpt>2</subscrpt> and <italic>u</italic><subscrpt>1</subscrpt> would have the ratio <italic>u</italic><subscrpt>2</subscrpt>:<italic>u</italic><subscrpt>1</subscrpt> = 2<supscrpt><italic>&kgr;</italic></supscrpt>, since <italic>u</italic> is identically zero. Accordingly, the expression (5) <italic>&kgr;</italic> = (log <italic>u</italic><subscrpt>2</subscrpt>:<italic>u</italic><subscrpt>1</subscrpt>)/log 2 gives an approximation for the asymptotic convergence exponent. In table I, α is the magnitude of the greatest angle divided by π ; <italic>n</italic> is the number of the interior points at which the values <italic>u</italic><subscrpt>1</subscrpt> and <italic>u</italic><subscrpt>2</subscrpt> are determined and then used to compute <italic>&kgr;</italic> from (5); <italic>&kgr;<subscrpt>m</subscrpt></italic> is the arithmetic mean of these <italic>n</italic> values and <italic>s</italic> the related quartile deviation; finally, <italic>&kgr;</italic> in the last column is the theoretical lower bound (6) <italic>&kgr</italic> = min (2, 1/&agr;α). In the investigation of problems with continuous and piecewise analytic boundary values the domains are similar to those previously described. Now, however, the limit solution for <italic>h</italic> → 0 is not known, and, therefore, the approximate values <italic>&kgr;</italic> are based on three consecutive approximations to the solution at the same point, with three different values of <italic>h</italic> which are chosen so that they are in the proportion 4:2:1. If these are denoted by <italic>u</italic><subscrpt>4</subscrpt>, <italic>u</italic><subscrpt>2</subscrpt>, and <italic>u</italic><subscrpt>1</subscrpt>, and if the truncation error were δ<subscrpt><italic>h</italic></subscrpt> = <italic>u<subscrpt>h</subscrpt></italic> - <italic>u</italic> = <italic>Ch<supscrpt>&kgr;</supscrpt></italic>, with <italic>C</italic> independent of <italic>h</italic>, then <italic>&kgr;</italic> could be computed from (7) <italic>&kgr;</italic> = (log <italic>u</italic><subscrpt>4</subscrpt> - <italic>u</italic><subscrpt>2</subscrpt>/<italic>u</italic><subscrpt>2</subscrpt> - <italic>u</italic><subscrpt>1</subscrpt>)/log 2. This is now the formula from which the approximate values <italic>&kgr;</italic> are computed at <italic>n</italic> (quite uniformly distributed) interior points. <italic>&kgr;<subscrpt>m</subscrpt></italic> is the arithmetic mean and <italic>s</italic> the quartile deviation. In addition to these values, table II gives also <italic>&kgr;</italic>, which is defined by (8) <italic>&kgr;</italic> = min (2, 2/α). The remarkably good coincidence between the corresponding values <italic>&kgr;<subscrpt>m</subscrpt></italic> and <italic>&kgr;</italic> is also indicated in figure 1; it contains, in addition to the experimental values <italic>&kgr;<subscrpt>m</subscrpt></italic>, also the graphs of the analytic expressions for <italic>&kgr;</italic> and <italic>&kgr;</italic>, from (6) and (8), respectively, with respect to 1/α. Those values which are theoretically proved to be lower bounds for the convergence exponent are represented as a solid line and those values in the hypothetical case as a broken one. In trying to prove theoretically the hypothesis that, for continuous boundary values, the relation <italic>&kgr;</italic> ≧ <italic>&kgr;</italic> is true, the decisive difficulty lies in finding proper estimates for the variation of the discrete solution function <italic>u<subscrpt>h</subscrpt></italic> in the vicinity of a corner. As long as the angle of this corner is less than π, i.e., α < 1, then the variation of <italic>u<subscrpt>h</subscrpt></italic> is of the order <italic>O</italic>(<italic>r</italic>), as shown in [4], <italic>r</italic> being the distance from the corner point. This agrees with the variation <italic>O</italic>(<italic>r</italic>) of the solution function <italic>u</italic> of Laplace's differential equation. Now, for α > 1, the variation of <italic>u</italic> can be proved to be of the order <italic>O</italic>(<i