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January  2021, 8(1): 33-58. doi: 10.3934/jcd.2021003

## The geometry of convergence in numerical analysis

Received  September 2019 Revised  June 2020 Published  August 2020

The domains of mesh functions are strict subsets of the underlying space of continuous independent variables. Spaces of partial maps between topological spaces admit topologies which do not depend on any metric. Such topologies geometrically generalize the usual numerical analysis definitions of convergence.

Citation: George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003
##### References:
 [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd edition, Addison-Wesley, 1978. [2] U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-algebraic Equations, SIAM, 1998. doi: 10.1137/1.9781611971392. [3] K. Back, Concepts of similarity for utility functions, J. Math. Econom., 15 (1986), 129-142.  doi: 10.1016/0304-4068(86)90004-2. [4] G. Beer, On the Fell topology, Set-valued Analysis, 1 (1993), 69-80.  doi: 10.1007/BF01039292. [5] G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8149-3. [6] G. Beer, A. Caserta, G. D. Maio and R. Lucchetti, Convergence of partial maps, J. Math. Anal. Appl., 419 (2014), 1274-1289.  doi: 10.1016/j.jmaa.2014.05.040. [7] R. D. Canary, D. B. A. Epstein and P. L. Green, Notes on notes of Thurston, in Fundamentals of Hyperbolic Geometry: Selected Expositions, London Math. Soc. Lecture Note Ser., 328, Cambridge Univ. Press, Cambridge, 2006. [8] C. Chabauty, Limite dénsembles et géométrie des nombres, Bull. Soc. Math. France, 78 (1950), 143-151. [9] C. Cuell and G. W. Patrick, Geometric discrete analogues of tangent bundles and constrained Lagrangian systems, J. Geom. Phys., 59 (2009), 976-997.  doi: 10.1016/j.geomphys.2009.04.005. [10] P. de la Harpe, Spaces of closed subgroups of locally compact groups, preprint, arXiv: 0807.2030. [11] J. M. G. Fell, A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc., 13 (1962), 472-476.  doi: 10.1090/S0002-9939-1962-0139135-6. [12] A. C. Hansen, A theoretical framework for backward error analysis on manifolds, J. Geom. Mech., 3 (2011), 81-111.  doi: 10.3934/jgm.2011.3.81. [13] A. Illanes and S. B. Nadler, Hyperspaces. Fundamentals and Recent Advances., Marcel Dekker, Inc., New York, 1999. [14] A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd edition, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2009. [15] A. Iserles, H. Z. Munthe-Kass, S. P. Norsett and A. Zanna, Lie-group methods, Acta Numerica, 9 (2000), 215-365.  doi: 10.1017/S0962492900002154. [16] K. Kuratowski, Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl. (4), 40 (1955), 61–67. doi: 10.1007/BF02416522. [17] K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. [18] K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Pańtwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968. [19] R. Lucchetti and A. Pasquale, A new approach to hyperspace theory, J. Convex Anal., 1 (1994), 173-193. [20] J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys., 199 (1998), 351-395.  doi: 10.1007/s002200050505. [21] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.  doi: 10.1017/S096249290100006X. [22] K. Matsuzaki, The Chabauty and the Thurston topologies on the hyperspace of closed subsets, J. Math. Soc. Japan, 69 (2017), 263-292.  doi: 10.2969/jmsj/06910263. [23] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc., 71 (1951), 152-182.  doi: 10.1090/S0002-9947-1951-0042109-4. [24] J. R. Munkres, Topology, Prentice Hall, 2000. [25] S. B. Nadler, Hyperspaces of Sets. A Text with Research Questions, Marcel Dekker, Inc., New York-Basel, 1978. [26] R. S. Palais, When proper maps are closed, Proc. Amer. Math. Soc., 24 (1970), 835-836.  doi: 10.2307/2037337. [27] G. W. Patrick and C. Cuell, Error analysis of variational integrators of unconstrained Lagrangian systems, Numerische Mathematik, 113 (2009), 243-264.  doi: 10.1007/s00211-009-0245-3. [28] W. Rudin, Functional Analysis, McGraw-Hill, 1973. [29] V. Runde, A Taste of Topology, Universitext, Springer, New York, 2005. [30] M. Schatzman, Numerical Analysis: A Mathematical Introduction, Clarendon Press, Oxford, 2002. [31] E. Süli and D. F. Mayer, An Introduction to Numerical Analysis, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511801181. [32] S. Willard, General Topology, Addison-Wesley, 1970.

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##### References:
 [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd edition, Addison-Wesley, 1978. [2] U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-algebraic Equations, SIAM, 1998. doi: 10.1137/1.9781611971392. [3] K. Back, Concepts of similarity for utility functions, J. Math. Econom., 15 (1986), 129-142.  doi: 10.1016/0304-4068(86)90004-2. [4] G. Beer, On the Fell topology, Set-valued Analysis, 1 (1993), 69-80.  doi: 10.1007/BF01039292. [5] G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8149-3. [6] G. Beer, A. Caserta, G. D. Maio and R. Lucchetti, Convergence of partial maps, J. Math. Anal. Appl., 419 (2014), 1274-1289.  doi: 10.1016/j.jmaa.2014.05.040. [7] R. D. Canary, D. B. A. Epstein and P. L. Green, Notes on notes of Thurston, in Fundamentals of Hyperbolic Geometry: Selected Expositions, London Math. Soc. Lecture Note Ser., 328, Cambridge Univ. Press, Cambridge, 2006. [8] C. Chabauty, Limite dénsembles et géométrie des nombres, Bull. Soc. Math. France, 78 (1950), 143-151. [9] C. Cuell and G. W. Patrick, Geometric discrete analogues of tangent bundles and constrained Lagrangian systems, J. Geom. Phys., 59 (2009), 976-997.  doi: 10.1016/j.geomphys.2009.04.005. [10] P. de la Harpe, Spaces of closed subgroups of locally compact groups, preprint, arXiv: 0807.2030. [11] J. M. G. Fell, A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc., 13 (1962), 472-476.  doi: 10.1090/S0002-9939-1962-0139135-6. [12] A. C. Hansen, A theoretical framework for backward error analysis on manifolds, J. Geom. Mech., 3 (2011), 81-111.  doi: 10.3934/jgm.2011.3.81. [13] A. Illanes and S. B. Nadler, Hyperspaces. Fundamentals and Recent Advances., Marcel Dekker, Inc., New York, 1999. [14] A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd edition, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2009. [15] A. Iserles, H. Z. Munthe-Kass, S. P. Norsett and A. Zanna, Lie-group methods, Acta Numerica, 9 (2000), 215-365.  doi: 10.1017/S0962492900002154. [16] K. Kuratowski, Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl. (4), 40 (1955), 61–67. doi: 10.1007/BF02416522. [17] K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. [18] K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Pańtwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968. [19] R. Lucchetti and A. Pasquale, A new approach to hyperspace theory, J. Convex Anal., 1 (1994), 173-193. [20] J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys., 199 (1998), 351-395.  doi: 10.1007/s002200050505. [21] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.  doi: 10.1017/S096249290100006X. [22] K. Matsuzaki, The Chabauty and the Thurston topologies on the hyperspace of closed subsets, J. Math. Soc. Japan, 69 (2017), 263-292.  doi: 10.2969/jmsj/06910263. [23] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc., 71 (1951), 152-182.  doi: 10.1090/S0002-9947-1951-0042109-4. [24] J. R. Munkres, Topology, Prentice Hall, 2000. [25] S. B. Nadler, Hyperspaces of Sets. A Text with Research Questions, Marcel Dekker, Inc., New York-Basel, 1978. [26] R. S. Palais, When proper maps are closed, Proc. Amer. Math. Soc., 24 (1970), 835-836.  doi: 10.2307/2037337. [27] G. W. Patrick and C. Cuell, Error analysis of variational integrators of unconstrained Lagrangian systems, Numerische Mathematik, 113 (2009), 243-264.  doi: 10.1007/s00211-009-0245-3. [28] W. Rudin, Functional Analysis, McGraw-Hill, 1973. [29] V. Runde, A Taste of Topology, Universitext, Springer, New York, 2005. [30] M. Schatzman, Numerical Analysis: A Mathematical Introduction, Clarendon Press, Oxford, 2002. [31] E. Süli and D. F. Mayer, An Introduction to Numerical Analysis, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511801181. [32] S. Willard, General Topology, Addison-Wesley, 1970.
Illustrating convergence in the lower and upper Vietoris topologies. Right: a subbasic neighbourhood of a subset $A$, in the upper Vietoris topology, is defined by an open set $U$. $A$ is contained in $U$ and the green sets are in the subbasic neighbourhood are contained in $U$. As $U$ shrinks, every point in the green sets in drawn to some point in $A$—everything approximable is in ${\rm cl}A$. At left, in the lower Vietoris topology, the green sets only have to intersect $U$, and shrinking $U$ around a single point of $A$ generates approximations when the green sets also meet $U$—everything in $A$ is approximable
Left: a neighbourhood of the geometric topology is defined by an open set $U$, a compact set $K$, and open sets $V_i$. Containment within $U$ has to occur only inside the compact set $K$, with effect that a convergent sequence of subsets $\langle A_i\rangle$ has $K\cap A_i$ finally contained in $U$, say for some $i\ge N$, but larger $K$ require larger $N$. As shown the set $A$ is inside the neighbourhood because its intersection with $K$ is contained in $U$ and contacts each $V_i$. Smaller $U$, and larger $K$, and more and smaller $V_i$, correspond to a smaller more restrictive neighbourhood. Right: a basic neighbourhood of a compact set $B$ in the geometric topology. A subset of $X$ is inside such a neighbourhood if it is contained in $U$ and contacts each $V_i$
Left: the discrete approximations $y_i$, $i = 1, 2, 3\ldots$ (circles) are limiting to a continuous $y$. Shown (squares on red curves) are three selections of subsequences from the graphs of $y_i$. Every such subsequence converges to the graph of $y$, and that graph is the limit of such subsequences. Right: an open neighbourhood of the red graph is defined by a compact set $K$, an open set $U$, and open sets $V_i$. $K$, which may be restricted to a product of compact sets, may be thought of as a frame within which proximity to the graph is controlled by $U$. The other black curves are in the neighbourhood because they also contact the $V_i$. Larger $K$, smaller $U$, and more and smaller $V_i$, correspond to smaller neighbourhoods
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