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

## The geometry of convergence in numerical analysis

 Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK S7N5E6, Canada

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.  Google Scholar [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.  Google Scholar [3] K. Back, Concepts of similarity for utility functions, J. Math. Econom., 15 (1986), 129-142.  doi: 10.1016/0304-4068(86)90004-2.  Google Scholar [4] G. Beer, On the Fell topology, Set-valued Analysis, 1 (1993), 69-80.  doi: 10.1007/BF01039292.  Google Scholar [5] G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8149-3.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] C. Chabauty, Limite dénsembles et géométrie des nombres, Bull. Soc. Math. France, 78 (1950), 143-151.   Google Scholar [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.  Google Scholar [10] P. de la Harpe, Spaces of closed subgroups of locally compact groups, preprint, arXiv: 0807.2030. Google Scholar [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.  Google Scholar [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.  Google Scholar [13] A. Illanes and S. B. Nadler, Hyperspaces. Fundamentals and Recent Advances., Marcel Dekker, Inc., New York, 1999.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] K. Kuratowski, Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl. (4), 40 (1955), 61–67. doi: 10.1007/BF02416522.  Google Scholar [17] K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966.  Google Scholar [18] K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Pańtwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968.  Google Scholar [19] R. Lucchetti and A. Pasquale, A new approach to hyperspace theory, J. Convex Anal., 1 (1994), 173-193.   Google Scholar [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.  Google Scholar [21] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.  Google Scholar [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.  Google Scholar [23] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc., 71 (1951), 152-182.  doi: 10.1090/S0002-9947-1951-0042109-4.  Google Scholar [24] J. R. Munkres, Topology, Prentice Hall, 2000.  Google Scholar [25] S. B. Nadler, Hyperspaces of Sets. A Text with Research Questions, Marcel Dekker, Inc., New York-Basel, 1978.  Google Scholar [26] R. S. Palais, When proper maps are closed, Proc. Amer. Math. Soc., 24 (1970), 835-836.  doi: 10.2307/2037337.  Google Scholar [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.  Google Scholar [28] W. Rudin, Functional Analysis, McGraw-Hill, 1973.  Google Scholar [29] V. Runde, A Taste of Topology, Universitext, Springer, New York, 2005.  Google Scholar [30] M. Schatzman, Numerical Analysis: A Mathematical Introduction, Clarendon Press, Oxford, 2002.   Google Scholar [31] E. Süli and D. F. Mayer, An Introduction to Numerical Analysis, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511801181.  Google Scholar [32] S. Willard, General Topology, Addison-Wesley, 1970.  Google Scholar

show all references

##### References:
 [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd edition, Addison-Wesley, 1978.  Google Scholar [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.  Google Scholar [3] K. Back, Concepts of similarity for utility functions, J. Math. Econom., 15 (1986), 129-142.  doi: 10.1016/0304-4068(86)90004-2.  Google Scholar [4] G. Beer, On the Fell topology, Set-valued Analysis, 1 (1993), 69-80.  doi: 10.1007/BF01039292.  Google Scholar [5] G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8149-3.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] C. Chabauty, Limite dénsembles et géométrie des nombres, Bull. Soc. Math. France, 78 (1950), 143-151.   Google Scholar [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.  Google Scholar [10] P. de la Harpe, Spaces of closed subgroups of locally compact groups, preprint, arXiv: 0807.2030. Google Scholar [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.  Google Scholar [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.  Google Scholar [13] A. Illanes and S. B. Nadler, Hyperspaces. Fundamentals and Recent Advances., Marcel Dekker, Inc., New York, 1999.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] K. Kuratowski, Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl. (4), 40 (1955), 61–67. doi: 10.1007/BF02416522.  Google Scholar [17] K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966.  Google Scholar [18] K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Pańtwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968.  Google Scholar [19] R. Lucchetti and A. Pasquale, A new approach to hyperspace theory, J. Convex Anal., 1 (1994), 173-193.   Google Scholar [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.  Google Scholar [21] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.  Google Scholar [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.  Google Scholar [23] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc., 71 (1951), 152-182.  doi: 10.1090/S0002-9947-1951-0042109-4.  Google Scholar [24] J. R. Munkres, Topology, Prentice Hall, 2000.  Google Scholar [25] S. B. Nadler, Hyperspaces of Sets. A Text with Research Questions, Marcel Dekker, Inc., New York-Basel, 1978.  Google Scholar [26] R. S. Palais, When proper maps are closed, Proc. Amer. Math. Soc., 24 (1970), 835-836.  doi: 10.2307/2037337.  Google Scholar [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.  Google Scholar [28] W. Rudin, Functional Analysis, McGraw-Hill, 1973.  Google Scholar [29] V. Runde, A Taste of Topology, Universitext, Springer, New York, 2005.  Google Scholar [30] M. Schatzman, Numerical Analysis: A Mathematical Introduction, Clarendon Press, Oxford, 2002.   Google Scholar [31] E. Süli and D. F. Mayer, An Introduction to Numerical Analysis, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511801181.  Google Scholar [32] S. Willard, General Topology, Addison-Wesley, 1970.  Google Scholar
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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