Michael Neilan
Professor
Department of Mathematics
University of Pittsburgh
Professor
Department of Mathematics
University of Pittsburgh
I am a professor in the Department of Mathematics, University of Pittsburgh. My research interests include computational PDEs with an emphasis on the finite element method. My recent work focuses on structure-preserving discretizations for incompressible fluid models and surface PDEs. Other interests include fully nonlinear PDEs (Monge-Ampere equations, optimal transport), and discontinuous Galerkin methods. I've written several survey articles that explain these topics in some detail:
On the divergence constraint in mixed finite element methods for incompressible flows. V. John, A.Linke, C.Merdon, M.Neilan, and L.G.Rebholz.
SIAM Review, 59(3):492-544, 2017 [preprint | article ]
The Stokes Complex: A review of exactly divergence-free finite element pairs for incompressible flows. M. Neilan.
75 Years of Mathematics of Computation, Contemporary Mathematics, volume 754 [article]
Numerical analysis of strongly nonlinear PDEs. M. Neilan, A.J. Salgado and W. Zhang.
Acta Numerica, 26:137-303, 2017 [preprint | article]
The Monge-Ampere equation. M. Neilan, A.J. Salgado, and W. Zhang.
Handbook of Numerical Analysis, vol 21:105-219, 2020 [preprint | chapter]
Recent developments in numerical methods for fully nonlinear second order partial differential equations. X. Feng, R. Glowinski, and M. Neilan.
SIAM Review, 55(2):205-267, 2013 [preprint | article]
I am also the co-organizer (co-ringmaster) of the Finite Element Circus, a biannual conference started in 1970 devoted to the theory and applications of the finite element method.
Mathematics of Computation (Associate Editor, 2017-2021; Editor, 2022-2023; Managing Editor, 2024-present)
IMA Journal of Numerical Analysis (Associate Editor, 2019-present)
Calcolo (Associate Editor, 2020-present)
Journal of Numerical Mathematics (Associate Editor, 2021-present)
Journal of Scientific Computing (Associate Editor, 2021-2023)
Advances in Applied Mathematics and Mechanics (Associate Editor, 2014-2018)
Hongzhi Wan (5th year)
David Poling (3rd year)
Yerim Kone (2nd year)
On the convergence of iterated penalty methods for structure-preserving discretizations of saddle point problems. P. Farrell, M. Neilan, C. Parker, and L.R. Scott
Submitted [preprint]
A TraceFEM C^0 interior penalty method for the surface biharmonic equation. M. Neilan and H. Wan
Submitted [preprint]
A local Fortin projection for the Scott--Vogelius elements on general meshes. F. Eickmann, J. Guzman, M. Neilan, L.R. Scott, and T. Tscherpel
Journal of Numerical Mathematics (to appear) [preprint]
An unfitted divergence-free higher order finite element method for the Stokes problem. M. Neilan, M. Olshanskii, and H. von Wahl
Submitted [preprint]
A Taylor-Hood finite element for the surface Stokes problem without penalization. A. Demlow and M. Neilan
SIAM Journal on Numerical Analysis, 64(2):565-600, 2026 [preprint | article]
A high-order, pressure-robust, and decoupled finite difference method for the Stokes problem. Q. Feng, B. Han, and M. Neilan
Mathematics and Computers in Simulation, 241: 634-649, 2026 [preprint | article]
A C^0 interior penalty method for the stream function formulation of the surface Stokes problem. M. Neian and H. Wan
ESAIM: Mathematical Modeling and Numerical Analysis, 59(2):1177-1211, 2025 [preprint | article]
A general degree divergence-free finite element method for the two-dimensional Stokes problem on smooth domains. R. Durst and M. Neilan
Journal of Scientific Computing 101(2), Paper No. 33, 2024 [preprint | article]
An Eulerian finite element method for the linearized Navier-Stokes problem in an evolving domain. M. Neilan and M. Olshanskii
IMA Journal of Numerical Analysis 44(6):3234-3258, 2024 [preprint | article]
A tangential and penalty-free finite element method for the surface Stokes problem. A. Demlow and M. Neilan
SIAM Journal on Numerical Analysis, 62(1):248-272, 2024 [preprint | article]
Discrete elasticity exact sequences on Worsey-Farin splits. S. Gong, J. Gopalakrishnan, J. Guzman, and M. Neilan
ESAIM: Mathematical Modeling and Numerical Analysis 57(6):3373-3402, 2023 [preprint | article]
Convergence of Lagrange finite elements for Maxwell eigenvalue problem in 3D. D. Boffi, S. Gong, J. Guzman, and M. Neilan
IMA Journal of Numerical Analysis 44(4):1911-1945, 2024 [preprint | article]
A note on the shape regularity of Worsey-Farin splits. S. Gong, J. Guzman, and M. Neilan
Journal of Scientific Computing 95(2), Paper No. 46, 2023 [preprint | article]
A stable and H1-conforming divergence-free finite element pair for the Stokes problem using isoparametric mappings. M. Neilan, and M.B. Otus
Calcolo 60(3), Paper No. 37, 2023 [article]
A CutFEM divergence-free discretization for the Stokes problem. H. Liu, M. Neilan, and M. Olshanskii
ESAIM: Mathematical Modeling and Numerical Analysis 57(1):143-165, 2023 [preprint | article]
A divergence-free finite element method for the Stokes problem with boundary correction. H. Liu, M. Neilan, and M.B. Otus
Journal of Numerical Mathematics 31(2):105-123, 2023 [preprint | article]
Exact sequences on Worsey-Farin splits. J. Guzman, A. Lischke and M. Neilan
Mathematics of Computation 91(338):2571-2608, 2022 [preprint | article]
The Scott-Vogelius method for the Stokes problem on anisotropic meshes. K. Kean, M. Neilan, and M. Schneier
International Journal of Numerical Analysis and Modeling 19(2-3):157-174, 2022 [preprint | article]
Low-order divergence-free approximations for the Stokes problem on Worsey-Farin and Powell-Sabin splits. M. Fabien, J. Guzman, M. Neilan, and A. Zytoon
Computer Methods in Applied Mechanics and Engineering 390, 2022 [preprint | article]
Convergence of Lagrange finite elements for the Maxwell eigenvalue problem in 2D. D. Boffi, J. Guzman, and M. Neilan
IMA Journal of Numerical Analysis, 43(2):663-691, 2023 [preprint | article]
Divergence-free Scott-Vogelius elements on curved domains. M. Neilan and M.B. Otus.
SIAM Journal on Numerical Analysis 59(2):1090-1116, 2021 [preprint | article]
Connection between grad-div stabilized Stokes finite elements and divergence-free elements. M. Neilan and A. Zytoon
International Journal of Numerical Analysis and Modeling 17(6):839-857, 2020 [article]
Pressure-robustness in quasi-optimal a priori estimates for the Stokes problem. A. Linke, C. Merdon, and M. Neilan
Electronic Transactions on Numerical Analysis 52:281-294, 2020 [preprint | article]
The Stokes Complex: A review of exactly divergence–free finite element pairs for incompressible flows. M. Neilan
75 Years of Mathematics of Computation, Contemporary Mathematics, volume 754, AMS [article]
Exact sequences of Powell-Sabin splits. J. Guzman, A. Lischke, and M. Neilan.
Calcolo, 57(13), 2020 [preprint | article]
The Monge-Ampere equation. M. Neilan, A.J. Salgado, and W. Zhang
Handbook of Numerical Analysis, vol 21:105-219, 2020 [preprint | chapter]
Exact smooth piecewise polynomial sequences on Alfeld splits. G. Fu, J. Guzman, and M. Neilan
Mathematics of Computation 89(323):1059-1091, 2020 [preprint | article]
Discrete Miranda-Talenti estimates and applications to linear and nonlinear PDEs. M. Neilan and M. Wu
Journal of Computational and Applied Mathematics, 356:358-376, 2019 [preprint | article]
Low-order Raviart-Thomas approximations of axisymmetric Darcy flow. M. Neilan and A. Zytoon
Journal of Mathematical Analysis and Applications 473(2):905-917, 2019 [preprint | article]
Rates of convergence in W2p-norm for the Monge-Ampere equation. M. Neilan and W. Zhang
SIAM Journal on Numerical Analysis 56(5):3099-3120, 2018 [preprint | article]
Inf-sup stable finite elements on barycentric refinements producing divergence-free approximations in arbitrary dimensions. J. Guzman and M. Neilan
SIAM Journal on Numerical Analysis, 56(5):2826-2844, 2018 [preprint | article]
Quasi-optimality of a pressure-robust nonconforming finite element method for the Stokes problem. A. Linke, C. Merdon, M. Neilan, and F. Neumann
Mathematics of Computation, 87(312):1543-1566, 2018 [preprint | article]
Interior penalty discontinuous Galerkin methods for second order linear non-divergence form elliptic PDEs. X. Feng, M. Neilan, and S. Schnake
Journal of Scientific Computing, 74(3):1651-1676, 2018 [preprint | article]
Macro Stokes elements on quadrilaterals. M. Neilan and D. Sap
International Journal of Numerical Analysis and Modeling, 15(4-5):729-745, 2018 [article]
Convergence analysis of a finite element method for second order non-variational elliptic problems. M. Neilan
Journal of Numerical Mathematics, 25(3):169-184, 2017 [article]
A connection between coupled penalty projection timestepping schemes with FE spatial discretization for the Naiver-Stokes equations. A. Linke, M. Neilan, L.G. Rebholz, and N.E. Wilson
Journal of Numerical Mathematics 25(4):229-248, 2017 [article]
On the divergence constraint in mixed finite element methods for incompressible flows. V. John, A. Linke, C. Merdon, M. Neilan, and L.G. Rebholz
SIAM Review 59(3):492-544, 2017 [preprint | article]
A posteriori estimates using auxiliary subspace techniques. H. Hakula, M. Neilan, and J.S. Ovall
Journal of Scientific Computing 72(1):97-127, 2017 [preprint | article]
Numerical analysis of strongly nonlinear PDEs. M. Neilan, A.J. Salgado and W. Zhang
Acta Numerica 26:137-303, 2017 [preprint | article]
A C0 interior penalty method for a von Karman plate. S.C. Brenner, M. Neilan, A. Reiser, and L.-Y. Sung
Numerische Mathematik 135(3):803-832, 2017 [preprint | article]
Finite element methods for second order linear elliptic partial differential equations in non-divergence form. X. Feng, L. Hennings, and M. Neilan
Mathematics of Computation 86(307):2025-2051, 2017 [article]
Dual-mixed finite element methods for the stationary Boussinesq problem. E. Colmenares and M. Neilan
Computers and Mathematics with Applications 72(7):1828-1850, 2016 [article]
Stable discontinuous Galerkin FEM without penalty parameters. L. John, M. Neilan, and I. Smears
Numerical Mathematics and Advanced Applications ENUMATH 2015.
Lecture Notes in Computational Science and Engineering, vol 112:165-174 [article]
A dual mixed finite element method for the Brinkman problem. J.S. Howell, M. Neilan, and N.J. Walkington
SMAI Journal of Computational Mathematics 2:1-17, 2016 [article]
DG finite element differential calculus and applications to numerical solutions of linear and nonlinear partial differential equations. X. Feng, M. Neilan, and T. Lewis
Journal of Computational and Applied Mathematics 299:68-91, 2016 [preprint | article]
Stokes elements on cubic meshes yielding divergence-free approximations. M. Neilan and D. Sap
Calcolo 53(3):263-283, 2016 [preprint | article]
Discrete and conforming smooth de Rham complexes in three dimensions. M. Neilan
Mathematics of Computation 84(295):2059-2081, 2015 [article]
Finite element methods for fully nonlinear second order PDEs based on the discrete Hessian. M. Neilan
Journal of Computational and Applied Mathematics 263:351-369, 2014 [article]
Finite element approximations of general fully nonlinear second order elliptic partial differential equations based on the vanishing moment method. M. Neilan and X. Feng
Computers and Mathematics with Applications 68(12):2182-2204, 2014 [article]
A C0 method for the biharmonic problem without extrinsic penalization. S.B.G. Karakoc and M. Neilan
Numerical Methods for Partial Differential Equations 30(4):1254-1278, 2014 [article]
Convergence analysis of a symmetric dual-wind discontinuous Galerkin method. M. Neilan and T. Lewis
Journal of Scientific Computing 59(3):602-625, 2014 [article]
Conforming and divergence-free Stokes elements in three dimensions. J. Guzman and M. Neilan
IMA Journal of Numerical Analysis 34(4):1489-1508, 2014 [article]
Symmetric and conforming mixed finite elements for plane elasticity using rational bubbles. J. Guzman and M. Neilan
Numerische Mathematik 126(1):153-171, 2014 [article]
Conforming and divergence-free Stokes elements on general triangular meshes. J. Guzman and M. Neilan
Mathematics of Computation, 83(285):15-36, 2014 [article]
Convergence analysis of a fourth order perturbation of the radially symmetric Monge-Ampere equation. X. Feng and M. Neilan
Applicable Analysis 93(8):1626-1646, 2014 [preprint | article]
A unified analysis for three finite element methods for the Monge-Ampere equation. M. Neilan
Electronic Transactions on Numerical Analysis 41:262-288, 2014 [article]
Quadratic finite element methods for the Monge-Ampere equation. M. Neilan
Journal of Scientific Computing 54(1):200-226, 2013 [article]
Stokes complexes and the construction of stable finite element methods with pointwise mass conservation. R. Falk and M. Neilan
SIAM Journal on Numerical Analysis 51(2):1308-1326, 2013 [article]
Recent developments in numerical methods for fully nonlinear second order partial differential equations. X. Feng, R. Glowinski, and M. Neilan
SIAM Review 55(2):205-267, 2013. [article]
Isoparametric C0 interior penalty methods for plate bending problems on smooth domains. S.C. Brenner, M. Neilan, and L.-Y. Sung
Calcolo 50(1):35-67, 2013 [article]
Finite element approximations of the three dimensional Monge-Ampere equation. S.C. Brenner and M. Neilan
ESAIM: Mathematical Modeling and Numerical Analysis, 46(5):979-1001, 2012 [article]
A family of non-conforming elements for the Brinkman problem. J. Guzman and M. Neilan
IMA Journal of Numerical Analysis 32(4):1484-1508, 2012 [article]
A family of non-conforming elements and the analysis of Nitsche’s method for a singularly perturbed fourth order problem. J. Guzman, D. Leykekhman, and M. Neilan
Calcolo 49(2):95-125, 2012 [article]
Error analysis of Galerkin approximations of the fully nonlinear Monge-Ampere equation. X. Feng and M. Neilan
Journal of Scientific Computing 47:303-327, 2011 [article]
C0 penalty methods for the fully nonlinear Monge-Ampere equation. S.C. Brenner, T. Gudi, M. Neilan, and L.-Y. Sung
Mathematics of Computation 80:1979-1995, 2011 [article]
C0 penalty methods for a fourth order elliptic singular perturbation problem. S.C. Brenner and M. Neilan
SIAM Journal on Numerical Analysis 49:869-892, 2011 [article]
An interior penalty method for a sixth order elliptic equation. T. Gudi and M. Neilan
IMA Journal of Numerical Analysis 31:1734-1753, 2011 [article]
Localized pointwise error estimates and global Lp error estimates of Nitsche’s method. M. Neilan
International Journal of Numerical Analysis, Series B 2(4):338-354, 2011 [article]
Discontinuous Galerkin methods for a bi-wave equation modeling d-wave superconductors. X. Feng and M. Neilan
Mathematics of Computation 30:1303-1333, 2011 [article]
A nonconforming Morley finite element method for the Monge-Ampere equation. M. Neilan
Numerische Mathematik 115(3):371-394, 2010 [article]
Finite element methods for a bi-wave equation modeling d-wave superconductors. X. Feng and M. Neilan
Journal of Computational Mathematics, 28(3):331-353, 2010 [article]
Finite element methods for a bi-wave equation modeling d-wave superconductors. X. Feng and M. Neilan
Journal of Computational Mathematics, 28(3):331-353, 2010 [preprint | article]
Mixed finite element approximations of the fully nonlinear Monge-Ampere equation based on the vanishing moment method. X. Feng and M. Neilan
SIAM Journal on Numerical Analysis, 47(2):1226-1250, 2009 [preprint | article]
Vanishing moment method and moment solutions for second order fully nonlinear partial differential equations. X. Feng and M. Neilan
Journal of Scientific Computing 38(1):74-98, 2009 [preprint | article]
Error analysis of finite element approximations of the inverse mean curvature flow arising from general relativity. X. Feng, M. Neilan, and A. Prohl
Numerische Mathematik 108(1):93-119, 2007 [article]