# American Institute of Mathematical Sciences

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, doi: 10.3934/jcd.2021003
##### References:

show all references

##### References:
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
 [1] Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112 [2] Enrico Gerlach, Charlampos Skokos. Comparing the efficiency of numerical techniques for the integration of variational equations. Conference Publications, 2011, 2011 (Special) : 475-484. doi: 10.3934/proc.2011.2011.475 [3] Cheng Wang. Convergence analysis of the numerical method for the primitive equations formulated in mean vorticity on a Cartesian grid. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1143-1172. doi: 10.3934/dcdsb.2004.4.1143 [4] Hongguang Xiao, Wen Tan, Dehua Xiang, Lifu Chen, Ning Li. A study of numerical integration based on Legendre polynomial and RLS algorithm. Numerical Algebra, Control & Optimization, 2017, 7 (4) : 457-464. doi: 10.3934/naco.2017028 [5] Ruijun Zhao, Yong-Tao Zhang, Shanqin Chen. Krylov implicit integration factor WENO method for SIR model with directed diffusion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4983-5001. doi: 10.3934/dcdsb.2019041 [6] Sebastián J. Ferraro, David Iglesias-Ponte, D. Martín de Diego. Numerical and geometric aspects of the nonholonomic SHAKE and RATTLE methods. Conference Publications, 2009, 2009 (Special) : 220-229. doi: 10.3934/proc.2009.2009.220 [7] Hao-Chun Lu. An efficient convexification method for solving generalized geometric problems. Journal of Industrial & Management Optimization, 2012, 8 (2) : 429-455. doi: 10.3934/jimo.2012.8.429 [8] Zhanyuan Hou. Geometric method for global stability of discrete population models. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3305-3334. doi: 10.3934/dcdsb.2020063 [9] Wen Li, Song Wang, Volker Rehbock. A 2nd-order one-point numerical integration scheme for fractional ordinary differential equations. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 273-287. doi: 10.3934/naco.2017018 [10] Michael Anderson, Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas and Michael Taylor. Metric tensor estimates, geometric convergence, and inverse boundary problems. Electronic Research Announcements, 2003, 9: 69-79. [11] Aurore Back, Emmanuel Frénod. Geometric two-scale convergence on manifold and applications to the Vlasov equation. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 223-241. doi: 10.3934/dcdss.2015.8.223 [12] Shuang Liu, Xinfeng Liu. Krylov implicit integration factor method for a class of stiff reaction-diffusion systems with moving boundaries. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 141-159. doi: 10.3934/dcdsb.2019176 [13] Antonia Katzouraki, Tania Stathaki. Intelligent traffic control on internet-like topologies - integration of graph principles to the classic Runge--Kutta method. Conference Publications, 2009, 2009 (Special) : 404-415. doi: 10.3934/proc.2009.2009.404 [14] Jinyan Fan, Jianyu Pan. On the convergence rate of the inexact Levenberg-Marquardt method. Journal of Industrial & Management Optimization, 2011, 7 (1) : 199-210. doi: 10.3934/jimo.2011.7.199 [15] Stefan Kindermann. Convergence of the gradient method for ill-posed problems. Inverse Problems & Imaging, 2017, 11 (4) : 703-720. doi: 10.3934/ipi.2017033 [16] Regina S. Burachik, C. Yalçın Kaya. An update rule and a convergence result for a penalty function method. Journal of Industrial & Management Optimization, 2007, 3 (2) : 381-398. doi: 10.3934/jimo.2007.3.381 [17] Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic & Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1 [18] Karl Kunisch, Markus Müller. Uniform convergence of the POD method and applications to optimal control. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4477-4501. doi: 10.3934/dcds.2015.35.4477 [19] Yong Duan, Jian-Guo Liu. Convergence analysis of the vortex blob method for the $b$-equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1995-2011. doi: 10.3934/dcds.2014.34.1995 [20] Jiequn Han, Jihao Long. Convergence of the deep BSDE method for coupled FBSDEs. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 5-. doi: 10.1186/s41546-020-00047-w

Impact Factor: