Publications
See also my profiles at Google Scholar + ResearchGate + arXiv.
Submitted for Publication
- S. Franz & N. Kopteva, Pointwise-in-time a posteriori error control for higher-order discretizations of time-fractional parabolic equations, November 2022, submitted for publication. See also arXiv:2211.06272.
- T. Linß, N. Kopteva, G. Radojev & M. Ossadnik, A review of maximum-norm a posteriori error bounds for time-semidiscretisations of parabolic equations, December 2022, submitted for publication. See also arXiv:2212.11540.
- N. Kopteva, Fully computable a posteriori error estimator using anisotropic flux equilibration on anisotropic meshes, submitted for publication.
Revised pdf version (June 2017). Pdf file (March 2017).
See also an extended pdf version (July 2017, with 2 appendices included): arXiv:1704.04404.
Edited Books
- 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.
Book Chapters
- 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.
Review Articles
- 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. Pdf file.
Refereed Articles
- A. Demlow, S. Franz & N. Kopteva, Maximum norm a posteriori error estimates for convection-diffusion problems, IMA J. Numer. Anal., (2022), accepted for publication. Revised pdf version (December 2022). See also arXiv:2207.08251.
- N. Kopteva & M. Stynes, A posteriori error analysis for variable-coefficient multiterm time-fractional subdiffusion equations, J. Sci. Comput., (2022), published online 14 July 2022; 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; 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; doi: 10.1016/j.aml.2021.107515.
Revised pdf version (July 2021). 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.
Revised pdf file (July 2020). Pdf file (May 2019). See also arXiv:1905.05070. - 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.
Revised pdf file (February 2020; additionally, an appendix is included which does not feature in the journal version). See also arXiv:1906.05703. - 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.
Revised pdf file (July 2020). Pdf file (January 2020). 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.
Revised pdf file (November 2019). Pdf file (May 2019). See also arXiv:1905.07426. - 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 (October 2018). See also arXiv:1810.09211. - 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.
Revised pdf file (October 2018). See also arXiv:1709.09136. - 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.
Revised pdf file (May 2019). Pdf file (November 2018). 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.
Revised pdf file (April 2018). 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, 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.
Revised pdf file (March 2017). Pdf file (May 2016) - 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 & 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. Revised pdf file (October 2016).
- 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.
Revised pdf file (July 2015). Pdf file (November 2014). - 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. Revised pdf file (June 2016). Pdf file (June 2015). - 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.
Journal file. Revised pdf file of May 2015 (more recent misprint corrections are highlighted in blue). Pdf file (August 2014). - 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. Revised pdf file (November 2014). Pdf file (June 2014). - 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.
Revised pdf file (January 2013). Pdf file (September 2012). - 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.
Final pdf file (March 2013). Revised pdf file (January 2013). Revised pdf file (May 2012).
(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 & 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. - 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. Revised pdf file (May 2011).
(An extended version of this paper is published in arXiv:2212.11916 on 22 Dec 2022.) - 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). Pdf file (December 2011). - 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. Revised pdf file (2011).
(Note: we have placed some proofs that contain many technical details in a separate paper arXiv:1004.1334.) - 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). Pdf file. - 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.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. Revised pdf file. Pdf file.
- 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.
- 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.) - 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.
- 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), 680-695. Revised 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.
- 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.
- 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.
- 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.
- 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, 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 & 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.
- 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.
- 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. Pdf file.
- 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, 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.
- 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, 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 & 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, 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.
- 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.
- 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.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, 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). - 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).
Non-Refereed Publications
- 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.
- 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.
Articles in Conference Proceedings
- 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.
Theses
- N.V. Kopteva, Uniform grid methods for certain singularly perturbed equations on layer-adapted grids, PhD Thesis, Moscow State University, Moscow, 1996 (118 pp., in Russian, pdf file).
Supervisor: Professor V.B. Andreev. - N.V. Kopteva, A Study of numerical methods for non-selfadjoint singularly pertubed equations, Diploma Thesis, Moscow State University, Moscow, 1993 (62 pp., in Russian).
Supervisor: Professor V.B. Andreev.
As a few people have expressed interest in the material of Chapter 4, here is Chapter 4 (pp. 16-21, in Russian, pdf file) and Summary of Chapter 4 in English (pdf file);
see also Title page and references (pdf file) , Introduction (pp. 2-6, pdf file) , Chapters 1-3 (pp. 7-16, pdf file) , Chapter 5 (pp. 22-35, pdf file) and Chapter 6 (pp. 36-61 without tables, pdf file).