Oleg Davydov Address University of Giessen Department of Mathematics Arndtstrasse 2 35392 Giessen Germany Email oleg.davydov@math.uni-giessen.de Homepage https://oleg-davydov.de University webpage https://www.uni-giessen.de/oleg.davydov German-Ukrainian Academic Society

Software

• mFDlab: Meshless Finite Difference Method

• BBFEM: Bernstein-Bézier Finite Elements

• TSFIT: Two-Stage Scattered Data Fitting

Research Topics

Recent Research Publications

• (with S. Solodky) On optimal recovery of unbounded operators from inaccurate data. Preprint, arXiv:2505.07123 https://arxiv.org/abs/2505.07123

• (with B. Degli Esposti) Meshless moment-free quadrature formulas arising from numerical differentiation. Preprint, arXiv:2409.03567 https://arxiv.org/abs/2409.03567

• (with C. Bracco, C. Giannelli and A. Sestini) A positive meshless finite difference scheme for scalar conservation laws with adaptive artificial viscosity driven by fault detection, Comput. Math. Appl., 190 (2025), 103-121. https://doi.org/10.1016/j.camwa.2025.04.006 (open access). Preprints (2024-2025): arXiv:2409.15544 https://arxiv.org/abs/2409.15544

• (with A. Westermann, A. Sokolov and S. Turek) Stability and accuracy of a meshless finite difference method for the Stokes equations, International Journal for Numerical Methods in Engineering, 126 (2025), e70000, https://doi.org/10.1002/nme.70000 (open access). Earlier version entitled "Numerical study of the RBF-FD method for the Stokes equations", Preprint: Eldorado - Repository of the TU Dortmund, Ergebnisbericht Nr. 662, 2023, https://eldorado.tu-dortmund.de/handle/2003/41366

• Overlap splines and meshless finite difference methods, in "Approximation Theory and Numerical Analysis Meet Algebra, Geometry, Topology: INdAM 2022" (Martina Lanini, Carla Manni and Henry Schenck, Eds.), pp.97-113, Springer INdAM Series, vol. 60, Springer, 2024. https://doi.org/10.1007/978-981-97-6508-9_5 Preprint: arXiv:2306.02391 https://arxiv.org/abs/2306.02391

• Error bounds for a least squares meshless finite difference method on closed manifolds, Advances in Computational Mathematics, 49 (2023), 48, https://doi.org/10.1007/s10444-023-10044-0 (open access). Preprints (2019-2023): arXiv:1910.03359 http://arxiv.org/abs/1910.03359

• (with D. T. Oanh and N. M. Tuong) Improved stencil selection for meshless finite difference methods in 3D, Journal of Computational and Applied Mathematics, 425 (2023), 115031. https://doi.org/10.1016/j.cam.2022.115031 Preprint: arXiv:2202.06426 http://arxiv.org/abs/2202.06426

• (with P. Suchde and T. Jacquemin) Point cloud generation for meshfree methods: An overview, Archives of Computational Methods in Engineering, 30 (2023), 889-915. https://doi.org/10.1007/s11831-022-09820-w (open access)

• (with W. P. Yeo) C1 piecewise quadratic hierarchical Bases, Applied and Computational Harmonic Analysis, 54 (2021), 250-272. Available online 23 March 2021. https://doi.org/10.1016/j.acha.2021.03.002

• Approximation with conditionally positive definite kernels on deficient sets, in "Approximation Theory XVI: Nashville 2019" (Gregory E. Fasshauer, Marian Neamtu and Larry L. Schumaker, Eds.), pp.27-38, Springer Proceedings in Mathematics & Statistics, vol. 336, Springer-Verlag, 2021. https://doi.org/10.1007/978-3-030-57464-2_3 Preprint version: arXiv:2006.13543 http://arxiv.org/abs/2006.13543

• (with M. Safarpoor) A meshless finite difference method for elliptic interface problems based on pivoted QR decomposition, Applied Numerical Mathematics, 161 (2021), 489-509. Available online 9 December 2020, https://doi.org/10.1016/j.apnum.2020.11.018

• (with M. Ainsworth and H. Wang), Some remarks on spectral convergence and stability of Iso-Geometric Analysis, Computer Methods in Applied Mechanics and Engineering, 372 (2020), 113408. https://doi.org/10.1016/j.cma.2020.113408

• (with O. Kozynenko and D. Skorokhodov) Optimal approximation order of piecewise constants on convex partitions, Journal of Complexity, 58 (2020), 101444. Published Online: 6 November 2019, https://doi.org/10.1016/j.jco.2019.101444, Preprint version: arXiv:1904.09005 http://arxiv.org/abs/1904.09005

• (with D. T. Oanh and N. M. Tuong) Octant-based stencil selection for meshless finite difference methods in 3D, Vietnam Journal of Mathematics, 48 (2020), 93-106. Published online 14 August 2019 http://link.springer.com/article/10.1007/s10013-019-00364-4 Springer Nature SharedIt view-only version: https://rdcu.be/bOTBL

• Selection of sparse sets of influence for meshless finite difference methods. Preprint (2019), arXiv:1908.01567 http://arxiv.org/abs/1908.01567

• (with A. Sokolov, D. Kuzmin, A. Westermann, and S. Turek) A flux-corrected RBF-FD method for convection dominated problems in domains and on manifolds, Journal of Numerical Mathematics, 27 (2019), 253-269. https://doi.org/10.1515/jnma-2018-0097. Preprint version: Eldorado - Repository of the TU Dortmund, Ergebnisbericht Nr. 587, 2018, https://eldorado.tu-dortmund.de/handle/2003/36905

• (with C. Bracco, C. Giannelli and A. Sestini) Fault and gradient fault detection and reconstruction from scattered data, CAGD, 75 (2019), 101786. https://doi.org/10.1016/j.cagd.2019.101786

• (with A. Sokolov and S. Turek) Numerical study of the RBF-FD level set based method for partial differential equations on evolving-in-time surfaces, in "Meshfree Methods for Partial Differential Equations IX" (Michael Griebel and Marc Alexander Schweitzer, Eds.), pp.117-134, Lecture Notes in Computational Science and Engineering, vol. 129, Springer-Verlag, 2019. https://doi.org/10.1007/978-3-030-15119-5_7 Preprint version: Eldorado - Repository of the TU Dortmund, Ergebnisbericht Nr. 579, 2017, https://eldorado.tu-dortmund.de/handle/2003/36231

• (with R. Schaback) Optimal stencils in Sobolev spaces, IMA J. Numer. Anal., 39 (2019), 398-422.

• (with C. Bracco, C. Giannelli, and A. Sestini) An application of numerical differentiation formulas to discontinuity curve detection from irregularly sampled data, Rend. Sem. Mat. Univ. Pol. Torino, 76 (2018), 69--76. http://www.seminariomatematico.polito.it/rendiconti/76-2/69.pdf Preprint version: arXiv:1812.07399

• (with R. Schaback) Minimal numerical differentiation formulas, Numer. Math., 140 (2018), 555-592.

• (with W. P. Yeo) Approximation by C1 splines on piecewise conic domains, in "Approximation Theory XV: San Antonio 2016" (Gregory E. Fasshauer and Larry L. Schumaker, Eds.), pp.21--37, Springer Proceedings in Mathematics & Statistics, vol. 201, Springer-Verlag, 2017.

• (with D. T. Oanh and H. X. Phu) Adaptive RBF-FD method for elliptic problems with point singularities in 2D, Applied Mathematics and Computation, 313 (2017), 474-497.

• (with A. Saeed) C1 quintic splines on domains enclosed by piecewise conics and numerical solution of fully nonlinear elliptic equations, Applied Numerical Mathematics, 116 (2017), 172-183.

• (with M. Buhmann) Error bounds for multiquadrics without added constants, J. Approx. Theory, 219 (2017), 1-14.

• (with M. Ainsworth and L. L. Schumaker) Bernstein-Bézier finite elements on tetrahedral-hexahedral-pyramidal partitions, Computer Methods in Applied Mechanics and Engineering, 304 (2016), 140-170.

• (with G. Kostin and A. Saeed) Polynomial finite element method for domains enclosed by piecewise conics, CAGD, 45 (2016), 48-72.

• (with R. Schaback) Error bounds for kernel-based numerical differentiation, Numer. Math., 132 (2016), 243-269.

• (with G. Andriamaro and J. Simkin) Implementation of enhanced shape functions for non-affine elements, IET Science, Measurement & Technology, 9 (2015), 184-188.

• (with M. Ainsworth and G. Andriamaro) A Bernstein-Bézier basis for arbitrary order Raviart-Thomas finite elements, Constr. Approx., 41 (2015), 1-22.

• (with F. Rabarison) Approximation by sums of piecewise linear polynomials, J. Approx. Theory, 185 (2014), 107-123.

• (with J. Prasiswa and U. Reif) Two-stage approximation methods with extended B-splines, Math. Comp., 83 (2014), 809-833.

• (with W. P. Yeo) Macro-element hierarchical Riesz bases, in "Mathematical Methods for Curves and Surfaces: 8th International Conference, Oslo, 2012" (M. Floater et al, Eds.), pp.112--134, LNCS 8177, Springer-Verlag, 2014.

• (with A. Saeed) Numerical solution of fully nonlinear elliptic equations by Böhmer's method, J. Comput. Appl. Math., 254 (2013), 43-54.

• (with W. P. Yeo) Refinable C2 Piecewise Quintic Polynomials on Powell-Sabin-12 Triangulations, J. Comput. Appl. Math., 240 (2013), 62-73.

• Approximation by piecewise constants on convex partitions, J. Approx. Theory, 164 (2012), 346-352.

• (with N. Ratnarajah, A. Simmons and A. Hojjatoleslami) A novel approach for improved tractography and quantitative analysis of probabilistic fibre tracking curves, Medical Image Analysis 16 (2012), 227-238.

• (with A. Saeed) Stable splitting of bivariate spline spaces by Bernstein-Bézier methods, in "Curves and Surfaces - 7th International Conference, Avignon, France, June 24-30, 2010" (J.-D. Boissonnat et al, Eds.), pp. 220-235, LNCS 6920, Springer-Verlag, 2012.

• (with M. Ainsworth and G. Andriamaro) Bernstein-Bézier finite elements of arbitrary order and optimal assembly procedures, SIAM J. Sci. Comp., 33 (2011), 3087-3109.

• (with D. T. Oanh) Computation of optimal shape parameter for multipoint Gaussian RBF-FD approximation of Poisson equation, University of Strathclyde Department of Mathematics and Statistics Research Report, 2011-12.

• (with D. T. Oanh) On the optimal shape parameter for Gaussian radial basis function finite difference approximation of the Poisson equation, Comput. Math. Appl., 62 (2011), 2143--2161.

• Algorithms and error bounds for multivariate piecewise constant approximation, in "Approximation Algorithms for Complex Systems," (E. H. Georgoulis, A. Iske and J. Levesley, Eds.), Springer Proceedings in Mathematics, Vol. 3, pp. 27--45, Springer-Verlag, 2011.

• (with D. T. Oanh) Adaptive meshless centres and RBF stencils for Poisson equation, J. Comput. Phys., 230 (2011), 287-304.

Publications 1995-2010

Former Doctoral Students