Capacity of the regular polyhedra. (English) Zbl 0698.51009
The paper contains new results on the capacitance of the five Platonic solids. Using a finite difference method, the author gives a numerical approximation for every regular polyhedron and compares this with results that are already known. (Therefore the paper is a good survey on this subject, too.)
In the cases of cube and tetrahedron, a method of estimating a posteriori the leading terms of the local truncation error (for the discrete method used) is developed, together with various techniques of manipulating these estimates to determine the influence of this error on that of the computed solution.
In the cases of cube and tetrahedron, a method of estimating a posteriori the leading terms of the local truncation error (for the discrete method used) is developed, together with various techniques of manipulating these estimates to determine the influence of this error on that of the computed solution.
Reviewer: H.Martini
MSC:
| 51M20 | Polyhedra and polytopes; regular figures, division of spaces |
| 49M25 | Discrete approximations in optimal control |
References:
| [1] | Ames, W. F., Numerical Methods for Partial Differential Equations (1977), Academic Press: Academic Press New York · Zbl 0219.35007 |
| [2] | Babu˛ska, I., Feedback, adaptivity, anda posteriori estimates in finite elements: aims, theory, and experience, (Babu˛ska, I.; Zienkiewicz, O. C.; Gago, J.; Oliveira, E. R. De A., Accuracy Estimates and Adaptive Refinements in Finite Element Computations (1986), Wiley: Wiley New York) |
| [3] | Bank, R. E., Analysis of a locala posteriori error estimate for elliptic equations, (Babu˛ska, I.; Zienkiewicz, O. C.; Gago, J.; Oliveira, E. R. de A., Accuracy Estimates and Adaptive Refinements in Finite Element Computations (1980), Wiley: Wiley New York) · Zbl 0569.65079 |
| [4] | Bank, R. E.; Weiser, A., Somea posteriori estimators for elliptical partial differential equations, Math. Comput., 44, 283-301 (1985) · Zbl 0569.65079 |
| [5] | Barnhill, R. E.; Wilcox, C. H., Computable error bounds for finite element approximations to the Dirichlet problem, Rocky Mount. J. Math., 12, 459-470 (1982) · Zbl 0503.65059 |
| [6] | Bouwkamp, C. J., A simple method of calculating electrostatic capacity, Physica, 24, 538-542 (1958) · Zbl 0085.20802 |
| [7] | Bramble, J. H.; Hubbard, B. E., A priori bounds on the discretization error in the numerical solution of the Dirichlet problem, Contrib. diff. Eqns., 2, 229-252 (1963) · Zbl 0196.50801 |
| [8] | Brown, C. S., A posteriori error analysis with applications to finite element methodology and finite difference methodology, (Dissertation (1989), Univ. Texas at Arlington: Univ. Texas at Arlington Tex.) |
| [9] | Brown, C. S., Software for the computation of the capacity of the polyhedra, (Technical Report 258 (1989), Univ. Texas at Arlington: Univ. Texas at Arlington Tex.) · Zbl 0698.51009 |
| [10] | Campbell, J. B., Finite difference techniques for ring capacitors, J. Engng Math., 9, 21-28 (1975) · Zbl 0352.65058 |
| [11] | Conlan, J.; Diaz, J. B.; Parr, W. E., On the capacity of the icosahedron, J. Math. Phys., 2, 259-261 (1961) · Zbl 0098.18502 |
| [12] | Coxeter, H. S.M., Regular Polytopes (1963), McMillan: McMillan New York · Zbl 0118.35902 |
| [13] | Diaz, J. B., Upper and lower bounds for quadratic integrals and, at a point, for solutions of linear boundary value problems, (Langer, R. E., Boundary Value Problems in Differential Equations (1960), Univ. of Wisconsin Press: Univ. of Wisconsin Press Madison, Wis.) · Zbl 0107.31102 |
| [14] | Duggan, R. C.; Godwin, P. J., On the capacity of regular polyhedra, IMA Jl appl. Math., 28, 283-286 (1982) · Zbl 0495.31002 |
| [15] | Fix, G. J.; Rose, M. E., A comparative study of finite element and finite difference methods, SIAM Jl numer. Analysis, 22, 250-261 (1985) · Zbl 0569.65077 |
| [16] | Fix, G. J.; Stephan, E., On finite element approximation to higher order elliptic systems, Arch rational Mech. Analysis, 91, 137-151 (1986) · Zbl 0597.65082 |
| [17] | Forsythe, G. E.; Wasow, W. R., Finite-difference Methods for Partial Differential Equations (1960), Wiley: Wiley New York · Zbl 0099.11103 |
| [18] | Fried, I., Condition of finite element matrices generated from nonuniform meshes, AIAA Jl, 10, 219-221 (1971) · Zbl 0242.65046 |
| [19] | Fried, I., The \(I_2\) and \(I_∞\) condition numbers …Conf. Brunel Univ., (Uxbridge (1972)) |
| [20] | Greenspan, D., Resolution of classical capacity problems by means of a digital computer, Can. J. Phys, 44, 2605-2614 (1966) |
| [21] | Greenspan, D., Discrete Numerical Methods in Physics and Engineering (1974), Academic Press: Academic Press New York · Zbl 0288.65001 |
| [22] | Greenspan, D.; Casulli, V., Numerical Analysis for Applied Mathematics, Science, and Engineering (1988), Addison-Wesley: Addison-Wesley New York · Zbl 0658.65001 |
| [23] | Greenspan, D.; Silverman, E., The calculation of electrostatic capacity by means of a high speed digital computer, (Proc. IEEE, 53 (1965)) · Zbl 0173.44603 |
| [24] | Holden, A., Shapes, Space and Symmetry (1977), Columbia Univ. Press: Columbia Univ. Press New York |
| [25] | Kenner, H., Geodesic Math and How to Use It (1976), Univ. of California Press: Univ. of California Press Berkeley |
| [26] | Lyusternik, L. A., Convex Figures and Polyhedra (1963), Dover Publications: Dover Publications New York · Zbl 0113.16201 |
| [27] | Mandelbaum, H.; Conte, S., Solid Geometry (1957), Ronald Press: Ronald Press New York · Zbl 0077.14003 |
| [28] | Mitchell, A. R.; Griffiths, D. F., The Finite Difference Method in Partial Differential Equations (1980), Wiley: Wiley New York · Zbl 0417.65048 |
| [29] | Nitsche, J.; Nitsche, J., Error estimates for the numerical solution of elliptic differential equations, Archs rational mech. Analysis, 4, 293-306 (1960) · Zbl 0097.33103 |
| [30] | Ochoa, C. M.; Fix, G. J.; Stingley, R. M.; Brown, C. S., Accuracy analysis for finite element modeling of thermomechanically loaded structures, AIAA Jl, 296-301 (April 1988) |
| [31] | Oliveira, E. R. de A., On the accuracy of finite difference solutions to elliptic partial differential equations, Int. J. numer. Meth. Engng, 21, 229-238 (1985) · Zbl 0565.65056 |
| [32] | Parr, W. E., Upper and lower bounds for the capacitance of the regular solids, SIAM Jl, 9, 334-386 (1961) · Zbl 0105.30001 |
| [33] | Payne, L. E.; Weinberger, H. F., New bounds in harmonic and biharmonic problems, J. Math. Phys., 33, 291-307 (1955) · Zbl 0064.09903 |
| [34] | Pearce, P.; Pearce, S., Polyhedra Primer (1978), Van Nostrand Reinhold: Van Nostrand Reinhold New York · Zbl 0438.52006 |
| [35] | Petrovsky, I. G., Lectures on Differential Equations (1954), Interscience: Interscience New York · Zbl 0059.08402 |
| [36] | Petrosvky, I. G., Partial Differential Equations (1967), Saunders: Saunders Philadelphia, Pa. · Zbl 0163.11706 |
| [37] | Po´lya, G.; Szego¨, G., Isoperimetric Inequalities in Mathematical Physics (1951), Princeton Univ. Press: Princeton Univ. Press Princeton, N.J. · Zbl 0044.38301 |
| [38] | Proter, M. H.; Weinberger, H. F., Maximum Principles in Differential Equations (1967), Prentice-Hall: Prentice-Hall Englewood Cliffs, N.J. · Zbl 0153.13602 |
| [39] | Pugh, A., Polyhedra: A Visual Approach (1976), Univ. of California Press: Univ. of California Press Berkeley · Zbl 0387.52006 |
| [40] | Robertson, S. A., Polytopes and Symmetry (1984), Cambridge Univ. Press · Zbl 0536.53059 |
| [41] | Rosen, J. B., Approximate solution and error bounds for quasi-linear elliptic boundary value problems, SIAM Jl numer. Analysis, 7, 80-102 (1970) · Zbl 0206.47501 |
| [42] | Rosser, J. B., Estimation of capacity with bounds for the error, I, (Technical Report 645 (1966), Univ. Wisconsin) · Zbl 0011.00201 |
| [43] | Strang, G.; Fix, G. J., An Analysis of the Finite Element Method (1973), Prentice-Hall: Prentice-Hall Englewood Cliffs, N.J. · Zbl 0278.65116 |
| [44] | Walsh, J. L.; Young, D., On the accuracy of the numerical solution of the Dirichlet problem by finite differences, J. Res. natn. Bur. Stand., 51, 343-363 (1953) · Zbl 0052.35204 |
| [45] | Wasow, W. R., On the truncation error in the solution of Laplace’s equation by finite differences, J. Res. natn. Stand., 48, 345-348 (1952) |
| [46] | Wasow, W. R., The accuracy of difference approximations to plane Dirichlet problems with piecewise analytic boundary values, Q. appl. Math., 15, 53-63 (1957) · Zbl 0077.11301 |
| [47] | Zienkiewicz, O. C., The Finite Element Method in Engineering Science (1971), McGraw-Hill: McGraw-Hill New York · Zbl 0237.73071 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
