Natalia Kopteva's publications

See also my profiles at Google Scholar + ResearchGate + arXiv + ORCID (identifier 0000-0001-7477-6926).

PUBLICATIONS in chronological order

By Topic:

Time-fractional subdiffusion equations

• N. Kopteva, Error analysis of an L2-type method on graded meshes for semilinear subdiffusion equations, Appl. Math. Lett., (2025), Volume 160, 109306 (open access); available online 10 September 2024; doi: 10.1016/j.aml.2024.109306.
  See also arXiv:2408.03420.
• S. Franz & N. Kopteva, On the solution existence for collocation discretizations of time-fractional subdiffusion equations, J. Sci. Comput., 100 (2024), article number 68 (open access); doi: 10.1007/s10915-024-02619-w.
  See also arXiv:2403.11847.
• S. Franz & N. Kopteva, Time stepping adaptation for subdiffusion problems with non-smooth right-hand sides, (2024).
  See also arXiv:2401.01586.
• S. Franz & N. Kopteva, Pointwise-in-time a posteriori error control for higher-order discretizations of time-fractional parabolic equations, J. Comput. Appl. Math., volume 427 (2023), 115122, published online 21-Feb-2023 (open access); doi: 10.1007/s10915-022-01936-2. See also arXiv:2211.06272.
• N. Kopteva & M. Stynes, A posteriori error analysis for variable-coefficient multiterm time-fractional subdiffusion equations, J. Sci. Comput., 92 (2022), article number 73; doi: 10.1007/s10915-022-01936-2. See also arXiv:2202.13357.
• N. Kopteva, Maximum principle for time-fractional parabolic equations with a reaction coefficient of arbitrary sign, Appl. Math. Lett., (2022), 108209 (open access); available online 27 May 2022; doi: 10.1016/j.aml.2022.108209.
  See also arXiv:2202.10220.
• N. Kopteva, Pointwise-in-time a posteriori error control for time-fractional parabolic equations, Appl. Math. Lett., 123 (2022), 107515 (open access); doi: 10.1016/j.aml.2021.107515.
  See also arXiv:2105.05848.
• N. Kopteva, Error analysis of an L2-type method on graded meshes for a fractional-order parabolic problem, Math. Comp., 90 (2021), 19-40; published electronically on 14-Jul-2020; doi: 10.1090/mcom/3552.
  Pdf file (UL repository). See also arXiv:1905.05070.
• N. Kopteva, Error analysis for time-fractional semilinear parabolic equations using upper and lower solutions, SIAM J. Numer. Anal., 58 (2020), 2212-2234; doi: 10.1137/20M1313015.
  See also arXiv:2001.04452.
• N. Kopteva & X. Meng, Error analysis for a fractional-derivative parabolic problem on quasi-graded meshes using barrier functions, SIAM J. Numer. Anal., 58 (2020), 1217-1238; doi: 10.1137/19M1300686.
  See also arXiv:1905.07426.
• N. Kopteva, Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions, Math. Comp., 88 (2019), 2135-2155; published electronically 23-Jan-2019; doi: 10.1090/mcom/3410.
  See also arXiv:1709.09136.

Space-fractional differential equations

• N. Kopteva & M. Stynes, Analysis and numerical solution of a Riemann-Liouville fractional derivative two-point boundary value problem, Adv. Comput. Math., 43 (2017), 77-99; doi: 10.1007/s10444-016-9476-x (published electronically 25-Aug-2016).
  Author sharing link.
• N. Kopteva & M. Stynes, An efficient collocation method for a Caputo two-point boundary value problem, BIT, 55 (2015), 1105-1123; doi:10.1007/s10543-014-0539-4 (published electronically 10-Dec-2014).
  Author sharing link. Pdf file (UL repository).

Finite elements on anisotropic meshes

• N. Kopteva, Lower a posteriori error estimates on anisotropic meshes, Numer. Math., 146 (2020) 159–179; doi: 10.1007/s00211-020-01137-9. Author sharing link.
  Pdf file (UL depository). See also arXiv:1906.05703 (additionally, an appendix is included which does not feature in the journal version).
• N. Kopteva, Improved energy-norm a posteriori error estimates for singularly perturbed reaction-diffusion problems on anisotropic meshes, in Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2018, Lect. Notes Comput. Sci. Eng., Springer (2020), 143-156; doi: org/10.1007/978-3-030-41800-7_9.
  Pdf file (UL repository). See also arXiv:1810.09211.
• N. Kopteva, Energy-norm a posteriori error estimates for singularly perturbed reaction-diffusion problems on anisotropic meshes, Numer. Math., 137 (2017), 607-642; doi: 10.1007/s00211-017-0889-3 (published electronically 2-May-2017). Author sharing link.
  
• N. Kopteva, Fully computable a posteriori error estimator using anisotropic flux equilibration on anisotropic meshes.
  See also an extended pdf version (July 2017, with 2 appendices included): arXiv:1704.04404.
• N. Kopteva, Energy-norm a posteriori error estimates for singularly perturbed reaction-diffusion problems on anisotropic meshes. Neumann boundary conditions, in in Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2016, Lect. Notes Comput. Sci. Eng., Springer, 2017, 141-154; doi: 10.1007/978-3-319-67202-1.
  Revised pdf file (March 2017). Pdf file (October 2016).
• N. Kopteva, Maximum-norm a posteriori error estimates for singularly perturbed reaction-diffusion problems on anisotropic meshes, SIAM J. Numer. Anal., 53 (2015), 2519-2544; doi: 10.1137/140983458.
  Revised pdf file of May 2015 (more recent misprint corrections are highlighted in blue). Pdf file (UL depository).
• N.V. Kopteva, The two-dimensional Sobolev inequality in the case of an arbitrary grid, Zh. Vychisl. Mat. Mat. Fiz., 38 (1998), no. 4, 596-599 (in Russian); translation in Comput. Math. Math. Phys., 38 (1998), no. 4, 574-577.
  Pdf file in English. Pdf file in Russian (journal version). Ps file in Russian (the author's version).

Counterexamples for finite element approximations

• N. Kopteva, How accurate are finite elements on anisotropic triangulations in the maximum norm?, J. Comput. Appl. Math., (2019), published electronically 8-Jul-2019; doi: 10.1016/j.cam.2019.06.032.
  See also arXiv:1811.05353.
• N. Kopteva, Logarithm cannot be removed in maximum norm error estimates for linear finite elements in 3D, Math. Comp., 88 (2019), 1527-1532; published electronically 28-Sep-2018; doi: 10.1090/mcom/3384.
  See also arXiv:1710.03262.
  
  NOTE: Reference [1] is to a less known paper by V. B. Andreev of 1989, which is available here.
• N. Kopteva, Linear finite elements may be only first-order pointwise accurate on anisotropic triangulations, Math. Comp., 83 (2014), 2061-2070; doi: 10.1090/S0025-5718-2014-02820-2.
  Pdf file (UL repository).
• N. Kopteva, How accurate is the streamline-diffusion FEM inside characteristic (boundary and interior) layers? Comput. Methods Appl. Mech. Engrg., 193 (2004), 4875-4889; doi: 10.1016/j.cma.2004.05.008. Pdf file (UL repository).

A posteriori error estimation

• T. Linß, N. Kopteva, G. Radojev & M. Ossadnik, A review of maximum-norm a posteriori error bounds for time-semidiscretisations of parabolic equations, Electron. Trans. Numer. Anal., 60 (2024), 99-122 (open access). See also arXiv:2212.11540.
• N. Kopteva & R. Rankin, Pointwise a posteriori error estimates for discontinuous Galerkin methods for singularly perturbed reaction-diffusion equations, SIAM J. Numer. Anal., 61 (2023), 1938-1961; doi: 10.1137/22M149733X; Sharable ePrint link.
• A. Demlow, S. Franz & N. Kopteva, Maximum norm a posteriori error estimates for convection-diffusion problems, IMA J. Numer. Anal., 43 (2023), 2562-2584 (open access); doi: 10.1093/imanum/drad001. PDF file (UL repository). See also arXiv:2207.08251.
• N. Kopteva & T. Linß, Improved maximum-norm a posteriori error estimates for linear and semilinear parabolic equations, Adv. Comput. Math., 43 (2017), 999-1022; doi: 10.1007/s10444-017-9514-3 (published electronically 2-Mar-2017). Author sharing link.
• A. Demlow & N. Kopteva, Maximum-norm a posteriori error estimates for singularly perturbed elliptic reaction-diffusion problems, Numer. Math., 133 (2016), 707-742; doi: 10.1007/s00211-015-0763-0 (published electronically 14-Aug-2015). Author sharing link.
  Pdf file (UL repository).
• N. Kopteva & T. Linß, Maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions, SIAM J. Numer. Anal., 51 (2013), 1494-1524; doi: 10.1137/110830563. Pdf file (UL repository).
  (Note that originally this article was presented in two parts:
  A posteriori error estimation for parabolic problems using elliptic reconstructions. I: Backward-Euler and Crank-Nicolson methods, Pdf file (April 2011);
  A posteriori error estimation for parabolic problems using elliptic reconstructions. II: A third-order discontinuous Galerkin method, Pdf file (October 2011).)
• N. Kopteva & T. Linß, Numerical study of maximum norm a posteriori error estimates for singularly perturbed parabolic problems, Lecture Notes in Comput. Sci., 8236 (2013), 50-61; doi: 10.1007/978-3-642-41515-9_5.
  Pdf file (November 2012).
• N. Kopteva & T. Linß, Maximum norm a posteriori error estimation for a time-dependent reaction-diffusion problem, Comput. Methods Appl. Math. 12 (2012), 189-205; doi: 10.2478/cmam-2012-0013.
  Revised pdf file (March 2012).
• N. Kopteva, Maximum norm a posteriori error estimate for a 2d singularly perturbed semilinear reaction-diffusion problem, SIAM J. Numer. Anal., 46 (2008), 1602-1618; doi: 10.1137/060677616. Revised pdf file.
• N.M. Chadha & N. Kopteva, Maximum norm a posteriori error estimate for a 3d singularly perturbed semilinear reaction-diffusion problem, Adv. Comput. Math., 35 (2011), 33-55; doi: 10.1007/s10444-010-9163-2 (published online 1 June 2010). Author sharing link.
• T. Linß & N. Kopteva, A posteriori error estimation for a defect-correction method applied to convection-diffusion problems, Int. J. Numer. Anal. Model., 7 (2010), no. 4, 718-733. Revised pdf file.

Green's funcions and asymptotic analysis for singularly perturbed differential equations

• S. Franz & N. Kopteva, Green's function estimates for a singularly perturbed convection-diffusion problem, J. Differential Equations, 252 (2012), 1521-1545; doi: 10.1016/j.jde.2011.07.033. Pdf file (UL repository).
  (An extended version of this paper is published in arXiv:2212.11916 on 22 Dec 2022.)
• S. Franz & N. Kopteva, On the sharpness of Green’s function estimates for a convection-diffusion problem, in M. Koleva & L. Vulkov, eds., Finite Difference Methods: Theory and Applications, Proceedings of the Fifth International Conference FDM: T&A'10, Lozenetz, Bulgaria, June 2010, Rousse University Press, Rousse, 2011, ISBN 978-954-8467-44-5, 44-57. Revised pdf file.
  (We also published this paper electronically in arXiv:1102.4520 on 22 Feb 2011.)
• S. Franz & N. Kopteva, Green's function estimates for a singularly perturbed convection-diffusion problem in three dimensions, Int. J. Numer. Anal. Model. Ser. B, 2 (2011), 124-141.
  Revised pdf file (May 2011).
• R.B. Kellogg & N. Kopteva, A singularly perturbed semilinear reaction-diffusion problem in a polygonal domain, J. Differential Equations, 248 (2010), 184–208; doi: 10.1016/j.jde.2009.08.020 (published online 11 September 2009). Revised pdf file
  (Note: we have placed some proofs that involve much computation in a separate paper arXiv:0902.0987.)
• R.B. Kellogg & N. Kopteva, Some asymptotic expansions for a semilinear reaction-diffusion problem in a sector, arXiv:0902.0987, published electronically 5 Feb 2009.
• S. Franz & N. Kopteva, Green's function estimates for a 2d singularly perturbed convection-diffusion problem: extended analysis, arXiv:2212.11916, published electronically 22 Dec 2022.
• S. Franz & N. Kopteva, Full analysis of the Green's function for a singularly perturbed convection-diffusion problem in three dimensions, arXiv:1103.2948, published electronically 15 Mar 2011.
• N. Kopteva & M. Stynes, Perturbed asymptotic expansions for interior-layer solutions of a semilinear reaction-diffusion problem with small diffusion, arXiv:1004.1334, published electronically 8 Apr 2010, last revised 21 Apr 2011.

Convergence of moving mesh methods

• N.M. Chadha & N. Kopteva, A robust grid equidistribution method for a one-dimensional singularly perturbed semilinear reaction-diffusion problem, IMA J. Numer. Anal., 31 (2011), 188-211; doi: 10.1093/imanum/drp033 (published online 11 August 2009). Revised pdf file.
• N. Kopteva, Maximum norm a posteriori error estimates for a one-dimensional singularly perturbed semilinear reaction-diffusion problem, IMA J. Numer. Anal., 27 (2007), 576-592; doi: 10.1093/imanum/drl020. Revised pdf file.
• N. Kopteva, Convergence theory of moving grid methods, in T. Tang & J. Xu, eds., Adaptive Computations: Theory and Algorithms, Science Press, Beijing, 2007, ISBN 978-7-03-018421-4 (see also CAM Digest ), 159-210. Pdf file.
• N. Kopteva, N. Madden & M. Stynes, Grid equidistribution for reaction-diffusion problems in one dimension, Numer. Algorithms, 40 (2005), no. 3, 305-322; doi: 10.1007/s11075-005-7079-6. Author sharing link.
• N. Kopteva & M. Stynes, A robust adaptive method for a quasilinear one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39 (2001), no. 4, 1446-1467; doi: 10.1137/S003614290138471X.
• N. Kopteva, Maximum norm a posteriori error estimates for a one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39 (2001), no. 2, 423-441; doi: 10.1137/S0036142900368642.

Error analysis for layer solutions on layer-adapted meshes

• N. Kopteva & E. O'Riordan, Shishkin meshes in the numerical solution of singularly perturbed differential equations, Int. J. Numer. Anal. Model., 7 (2010), no. 3, 393-415. Journal pdf file.
• A.F. Hegarty, N. Kopteva, E. O'Riordan & M. Stynes, eds., BAIL 2008 - Boundary and Interior Layers, Proceedings of the International Conference on Boundary and Interior Layers - Computational and Asymptotic Methods, Limerick, July 2008, Lect. Notes Comput. Sci. Eng., 69, Springer, Berlin Heidelberg, 2009, ISBN 978-3-642-00604-3.
• N. Kopteva & M. Pickett, A second-order overlapping Schwarz method for a 2d singularly perturbed semilinear reaction-diffusion problem, Math. Comp. 81 (2012), 81-105; doi: doi.org/10.1090/S0025-5718-2011-02521-4 (published online 18 July 2011).
  Revised pdf file (Nov-2010; most recent misprint corrections are highlighted in blue).
  Pdf file (UL repository).
• N. Kopteva & M. Stynes, Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction-diffusion problem, Numer. Math., 119 (2011), 787-810; doi: 10.1007/s00211-011-0395-y (published online 15 July 2011). Author sharing link. Pdf file (UL repository).
  (Note: we have placed some proofs that contain many technical details in a separate paper arXiv:1004.1334.)
• N. Kopteva & S.B. Savescu, Pointwise error estimates for a singularly perturbed time-dependent semilinear reaction-diffusion problem, IMA J. Numer. Anal., 31 (2011), 616-639; doi: 10.1093/imanum/drp032 (published online 8 January 2010). Revised pdf file. Pdf file.
• N. Kopteva, M. Pickett & H. Purtill, A robust overlapping Schwarz method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions, Int. J. Numer. Anal. Model., 6 (2009), no. 4, 680-695. Journal pdf file.
• N. Kopteva, Numerical analysis of a 2d singularly perturbed semilinear reaction-diffusion problem, Lecture Notes in Comput. Sci., 5434 (2009), 80-91; doi: 10.1007/978-3-642-00464-3_8. Pdf file.
• V.B. Andreev & N. Kopteva, Pointwise approximation of corner singularities for a singularly perturbed reaction-diffusion equation in an L-shaped domain, Math. Comp., 77 (2008), 2125-2139; doi: 10.1090/S0025-5718-08-02106-6. Revised pdf file.
• N. Kopteva, Maximum norm error analysis of a 2d singularly perturbed semilinear reaction-diffusion problem, Math. Comp., 76 (2007), 631-646; doi: 10.1090/S0025-5718-06-01938-7.
  Revised pdf file (most recent misprint corrections are highlighted in blue).
• N. Kopteva & M. Stynes, Numerical analysis of singularly perturbed nonlinear reaction-diffusion problems with multiple solutions, Comput. Math. Appl., 51 (2006), 857-864; doi: 10.1016/j.camwa.2006.03.013.
• N. Kopteva, Error analysis of a 2d singularly perturbed semilinear reaction-diffusion problem, in R. Ciegis, ed., Mathematical Modelling and Analysis, 2005, Technika, Vilnius, 2005, 227-233. Pdf file.
• N. Kopteva & M. Stynes, Numerical analysis of a singularly perturbed nonlinear reaction-diffusion problem with multiple solutions, Appl. Numer. Math., 51 (2004), 273-288; doi: 10.1016/j.apnum.2004.07.001. Pdf file.
• N. Kopteva, Error expansion for an upwind scheme applied to a two-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 41 (2003), no. 5, 1851-1869; doi: 10.1137/S003614290241074X. Ps file.
• N. Kopteva & M. Stynes, Approximation of derivatives in a convection-diffusion two-point boundary value problem, Appl. Numer. Math., 39 (2001), no. 1, 47-60; doi: 10.1016/S0168-9274(01)00051-4.
• N. Kopteva, Uniform pointwise convergence of difference schemes for convection-diffusion problems on layer-adapted meshes, Computing, 66 (2001), no. 2, 179-197; doi: 10.1007/s006070170034. Author sharing link. Pdf file (UL repository).
• N. Kopteva & T. Linß, Uniform second order pointwise convergence of a central difference approximation for a quasilinear convection-diffusion problem, J. Comput. Appl. Math., 137 (2001), no. 2, 257-267; doi: 10.1016/S0377-0427(01)00353-3.
• N. Kopteva, Pointwise error estimates for 2d singularly perturbed semilinear reaction-diffusion problems, in I. Farago, P. Vabishchevich & L. Vulkov, eds., Finite Difference Methods: Theory and Applications, Proceedings of the 4th International Conference, Lozenetz, Bulgaria, August 2006, 105-114. Pdf file.
• N. Kopteva & M. Stynes, Approximation of interior-layer solutions in a singularly perturbed nonlinear reaction-diffusion problem, in A. R. Ansari, A. F. Hegarty & G. I. Shishkin, eds., Numerical Methods for Problems with Layer Phenomena, Proceedings of the 3rd Annual Workshop, Department of Mathematics & Statistics, University of Limerick, Ireland, February 2004, 20-25.
• N. Kopteva, N. Madden & M. Stynes, On equidistribution for reaction-diffusion problems, in A. R. Ansari, A. F. Hegarty & G. I. Shishkin, eds., Numerical Methods for Problems with Layer Phenomena, Proceedings of the 3rd Annual Workshop, Department of Mathematics & Statistics, University of Limerick, Ireland, February 2004, 46-51.
• V.B. Andreev & N.V. Kopteva, Uniform with respect to a small parameter convergence of difference schemes for a convection-diffusion problem, in J. J. H. Miller, G. I. Shishkin & L. Vulkov, eds., Analytical and Computational Methods for Convection-Dominated and Singularly Perturbed Problems, Nova Science Publishers, New York, 2000, 133-139. Ps file.
• N.V. Kopteva, On the convergence, uniform with respect to the small parameter, of a scheme with central difference on refined grids, Zh. Vychisl. Mat. Mat. Fiz., 39 (1999), no. 10, 1662-1678 (in Russian); translation in Comput. Math. Math. Phys., 39 (1999), no. 10, 1594-1610.
  Pdf file in English. Pdf file in Russian (journal version). Ps file in Russian (the author's version).
• V.B. Andreev & N.V. Kopteva, On the convergence, uniform with respect to a small parameter, of monotone three-point difference schemes, Differ. Uravn., 34 (1998), no. 7, 921-928 (in Russian); translation in Differential Equations, 34 (1998), no. 7, 921-929 (1999).
  Ps file in Russian. Scanned pdf file in Russian. Scanned pdf file in English.
• N.V. Kopteva, On the convergence, uniform with respect to a small parameter, of a scheme with weights for a one-dimensional nonstationary convection-diffusion equation, Zh. Vychisl. Mat. Mat. Fiz., 37 (1997), no. 10, 1213-1220 (in Russian); translation in Comput. Math. Math. Phys., 37 (1997), no. 10, 1173-1180.
  Pdf file in English. Pdf file in Russian (journal version). Ps file in Russian (the author's version).
• N.V. Kopteva, On the convergence, uniform with respect to the small parameter, of a difference scheme for an elliptic problem in a strip, Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet., (1997), no. 2, 6-9 (in Russian); translation in Moscow Univ. Comput. Math. Cybernet., (1997), no. 2, 7-12.
  Ps file in Russian (the author's version). Pdf file in Russian (journal version). Pdf file in English.
• N.V. Kopteva, On the convergence, uniform with respect to a small parameter, of a four-point scheme for a one-dimensional stationary convection-diffusion equation, Differ. Uravn., 32 (1996), no. 7, 951-957 (in Russian); translation in Differential Equations, 32 (1996), no. 7, 958-964.
  Ps file in Russian (the author's version). Pdf file in English (journal version).
• V.B. Andreev & N.V. Kopteva, Investigation of difference schemes with an approximation of the first derivative by a central difference relation, Zh. Vychisl. Mat. i Mat. Fiz., 36 (1996), no. 8, 101-117 (in Russian); translation in Comput. Math. Math. Phys., 36 (1996), no. 8, 1065-1078.
  Pdf file in English. Pdf file in Russian (journal version). Ps file in Russian (the authors' version).
  

Applied problems

• E.S. Benilov, M.S. Benilov & N. Kopteva, Steady rimming flows with surface tension, J. Fluid Mech., 597 (2008), 91-118; doi: 10.1017/S0022112007009585.
• A.C. Fowler, N. Kopteva & C. Oakley, The formation of river channels, SIAM J. Appl. Math., 67 (2007), 1016-1040; doi: 10.1137/050629264. Pdf file (UL repository).
• E.S. Benilov, N. Kopteva & S.B.G. O'Brien, Does surface tension stabilise a liquid film inside a rotating horizontal cylinder? Quart. J. Mech. Appl. Math., 58 (2005), 185-200; doi: 10.1093/qjmamj/hbi004.