Robert Kirby

Current Contact Information

Department Mathematics
Baylor University
Sid Richardson Building, Room 334B
Phone: (254)710-4846
Fax: (254)710-3569
Email: reduce(operator.add , [“Robert”,”_”,”Kirby”,”@”,”baylor”,”.”,”edu”], “”)

Previous Positions

Associate Professor, Baylor University, 2012-2016.
Associate Professor, Texas Tech University, 2006-2012.
Assistant Professor, University of Chicago, 2002-2006.
Dickson Instructor, University of Chicago, 2000-2002.
Ph.D., University of Texas at Austin, 2000.

Research Interests

Computers were invented to automate tedious and error-prone tasks, like the vast hoards of arithemtic operations required to perform advanced numerical simulations of science and engineering problems. However, programming computers is itself a tedious and error-prone task. So, why not get a computer to do it?

At the intersection of mathematics and computer science, one finds “metanumerical computing” – the use of mathematical structure to generate, manipulate, and optimize numerical software. I have contributed to software projects such as the FEniCS project, and Firedrake.  The goal is to fuse together aspects of domain-specific languages with structural and algorithmic aspects of finite elements to produce easy-to-use yet highly efficient code systems that provide efficient implementations of state-of-the-art numerical methods. Or, you can call it “numerical methods with a universal quantifier”.

These are under active development and yet widely used in applications.  I am also interested in pressing forward basic research in algorithms and finite element analysis. One ongoing project is to develop low-complexity simplicial finite element methods based on Bernstein polynomials.  These allow calculations on unstructured meshes with the same complexity as tensor-product elements, and also admit a wide range of different approximating spaces.  How we do present such structured calculations to high-level finite element code?  This problem is tackled in FInAT, which, unlike FIAT, is not a tabulator.

Also, given the ability to solve one problem well, how do we solve two problems glued together?  Multiphysics problems, both from the standpoint of theoretical analysis and the development of efficient preconditioners, are one way I utilize the ability of codes such as FEniCS and Firedrake.

Given the ability to efficiently produce efficient simulation codes, how can we turn back toward a deeper understanding of numerical methods and their application?  On this front, I have an ongoing interest in wave equations such as shallow water.  In addition to traditional stability and error estimates, how can we understand energy conservation or dissipation in the presence of damping?  How closely do numerical methods track physical expectations?  How can we use the kinds of properties to get more refined estimates?


My students should check Canvas as the semester begins for course information.



  • R. C. Kirby, “On the convergence of monolithic multigrid for implicit Runge-Kutta time stepping of finite element problems,” arXiv:2304.1879.  To appear, SIAM J. Scientific Computing.
  • R. C. Kirby and D. Shapero, “High-order bounds-satisfying approximation of partial differential equations via finite element variational inequalities,” arXiv:2311.05880.  To appear, Numerische Mathematik.
  • M. Homolya, R. C. Kirby, and D. A. Ham, “Exposing and exploiting structure: optimal code generation for high-order finite element methods,” arXiv:1711.02473.


  • R. C. Kirby, A. Klöckner, and X. Wei, “Exact domain truncation for the Morse-Ingard equations,” Computers & Mathematics with Applications 162(15): 22-32 (2024) arXiv:2210.13765.
  • A. Bock, C. J. Cotter, and R. C. Kirby, “Planar curve registration using Bayesian inversion,”   Computers & Mathematics with Applications 159: 155-172 (2024) (link) (arXiv:2307.04909).
  • X. Wei, A. Klöckner, and R. C. Kirby, “Integral equation methods for the Morse-Ingard equations,” Journal of Computational Physics, 112416 (2023) (arXiv:2210.12542).
  • C. J. Cotter, R. C. Kirby, and H. Morris, “Weighted-norm preconditioners for a multi-layer tide model,” SIAM J. Scientific Computing 45(4): A1789-A1811 (2023) (arXiv:2207.02116).
  • J. Marchena-Menendez and R. C. Kirby, “Additive Schwarz methods for serendipity elements,” SIAM J. Scientific Computing 45(3): S401-S420 (2023) (arXiv:2104.11842).
  • F. R. A. Aznaran, P. E. Farrell, and R. C. Kirby, “Transformations for Piola-mapped elements,” SMAI J. Computational Mathematics 8: 399-438 (2022) (arXiv:2110.13224).
  • K. J. Roberts, A. Olender, L. Franceschini, R. C. Kirby, R. S. Gioria, and B. S. Carmo, “Full waveform inversion using triangular waveform adapted meshes,” Geoscientific Model Development 15(23), 8639-8667 (2022) (arXiv:2108.10946).
  • L. Allen and R. C. Kirby, “Bounds-constrained polynomial approximation using the Bernstein basis,”  Numerische Mathematik 151(1): 101–126 (2022) (arXiv:2104.11819)
  • J. Crum, C. Cheng, D. A. Ham, L. Mitchell, R. C. Kirby, J. A. Levine, and A. Gillette, “Bringing Trimmed Serendipity Methods to Computational Practice in Firedrake,” ACM Trans. Math. Software 48(1): 1–19 (2022) (arXiv:2104.12986).
  • P. E. Farrell, R. C. Kirby, and J. Marchena-Menendez, “Irksome: Automating Runge-Kutta time-stepping for finite element methods,” .  ACM Trans. Math. Software 47(4):1–26 (2021) (arXiv:2006.16282).
  • R. C. Kirby, A. Klöckner, and B. Sepanski, “Finite elements for Helmholtz equations with a nonlocal boundary condition,”   SIAM J. Scientific Computing 43(3): A1671–A1691 (2021)(arXiv:2009.08493).
  • L. Allen and R. C. Kirby, “Structured inversion of the Bernstein-Vandermonde matrix,” SIAM J. Matrix Analysis and Applications 42(2): 557-577 (2021) (arXiv:2005.03251).
  • O. T. Kernell and R. C. Kirby, “Preconditioning mixed finite elements for tide models,” Computers and Mathematics with Applications 82: 212–227 (2021) (arXiv:2003.01632).
  • L. Allen and R. C. Kirby, “Structured inversion of the Bernstein mass matrix,” SIAM J. Matrix Analysis and Applications 41(2): 413–431 (2020) (arXiv:1907.05773).
  • P. Coogan and R. C. Kirby, “Optimal-order preconditioners for the Morse-Ingard equations,” Computers and Mathematics with Applications 79(8): 4258–2471 (2020). (arXiv:1911.10247)
  • R. C. Kirby and L. Mitchell, “Code generation for generally mapped finite elements,” ACM Trans. Math. Software 45(5): 1–23 (2019). (link) (arXiv:1808.05513).
  • C. J. Cotter, P. J. Graber, and R. C. Kirby, “Mixed finite elements for global tides with nonlinear damping,” Numerische Mathematik 140(4): 963–991 (2018). (link) (arXiv:1706.01353).
  • R. C. Kirby, “A general approach to transforming finite elements,” SMAI Journal of Computational Mathematics 4: 197-224 (2018). (link) (arXiv:1706.09017).
  • R. C. Kirby and L. Mitchell, “Solver composition across the PDE/linear algebra barrier,” SIAM J. Scientific Computing 40(1): C76–C98 (2018).  (pdf) (arXiv:1706.01356).
  • R. C. Kirby, “Fast inversion of the simplicial Bernstein mass matrix,” Numerische Mathematik 135:73–95 (2017) (arXiv: 1504.03990).
  • C. J. Cotter and R. C. Kirby, “Mixed finite elements for global tide models,” Numerische Mathematik 133(2): 255–277 (2016) (pdf).
  • B. W. Brennan and R. C. Kirby, “Finite element approximation and preconditioners for a coupled thermal-acoustic model, Computers and Mathematics with Applications 70(10): 2324 (2015)
  • R. C. Kirby and T. T. Kieu, “Symplectic-mixed finite element approximation of linear wave equations,” Numerische Mathematik 130:257 — 291 (2015), DOI 10.1007/s00211-014-0667-4. (pdf)
  • R. C. Kirby, “High-performance evaluation of finite element variational forms via commuting diagrams and duality,” ACM Trans. Math. Software 40(4):25:1–25:24 (2014) (pdf).
  • C. Liaw, W. King, and R.C. Kirby, “Delocalization for the 3-D discrete random Schroedinger operator at weak disorder,” J. Phys. A: Math. Theor. 47(2014) (pdf) (arXiv).
  • R. C. Kirby, “Low-complexity finite element algorithms for the de Rham complex on simplices,” SIAM J. Scientific Computing 36(2): A846–A868 (2014) (pdf).
  • B. Brennan, R. C. Kirby, J. Zweck, and S. Minkoff, “High-performance Python-based simulations of trace gas sensors,” PyHPC 2013. (pdf)
  • V. Howle, R. Kirby, and G. Dillon, “Block Preconditioners for Coupled Physics Problems”, SIAM J. Scientific Computing 35(5): S368–S385 (2013). (pdf)
  • B. Brennan, V. E. Howle, K. Kennedy, R. C. Kirby, and K. R. Long, “Playa: High-performance programmable linear algebra,” Scientific Programming 20(3): 257 — 273 (2012). (pdf)
  • P. Bochev, H. C. Edwards, R. C. Kirby, K. Peterson, and D. Ridzal, “Solving PDEs with Intrepid,” Scientific Programming 20(2): 151 — 180 (2012). (pdf)
  • R. C. Kirby, and T. T. Kieu, “Fast simplicial quadrature-based finite element operators using Bernstein polynomials,” Numerische Mathematik 121(2): 261 — 279 (2012). (pdf)
  • V. E. Howle and R. C. Kirby, “Block preconditioners for finite element discretization of incompressible flow with thermal convection,” Numerical Linear Algebra with Applications 19(2): 427 — 440 (2012). (pdf)
  • R. C. Kirby, “Fast simplicial finite element algorithms using Bernstein polynomials,” Numerische Mathematik 117(4): 631 — 652 (2011). (pdf)
  • K. R. Long, R. C. Kirby, and B. van Bloemen Waanders, “Unified embedded parallel finite element computations via software-based Frechet differentiation,” SIAM J. Scientific Computing 32(6):3323 — 3351 (2010). (pdf)
  • R. C. Kirby, “From functional analysis to iterative methods,” SIAM Review 52(2): 269 — 293 (2010). (pdf)
  • R. C. Kirby, “Singularity-free evaluation of collapsed-coordinate orthogonal polynomials,” ACM Trans. Math Software 37(1): 1 — 16 (2010). (pdf)
  • M. E. Rognes, R. C. Kirby, and A. Logg, “Efficient assembly of H(div) and H(curl) conforming finite elements,” SIAM J. Scientific Computing 31(6):4130–4151 (2009). (pdf)
  • A. R. Terrell, L. R. Scott, M. G. Knepley and R. C. Kirby, “Automated FEM Discretizations of the Stokes equations,” BIT Numerical Mathematics, 48(2):389–404 (2008). (pdf)
  • R. C. Kirby and A. Logg, “Benchmarking domain-specific compiler optimizations for variational forms,” ACM Trans. Math. Software 35(2):1–18 (2008). (pdf)
  • R. C. Kirby and L. R. Scott, “Geometric optimization of the evaluation of finite element operators”, SIAM J. Scientific Computing 29:827–841 (2007). (pdf)
  • R. C. Kirby and A. Logg, “Efficient compilation of a class of variational forms”, ACM Trans. Math. Software 33(3):1– 20 (2007). (pdf)
  • R. C. Kirby and A. Logg, “A compiler for variational forms,” ACM Trans. Math. Software. 32:417-444 (2006). (pdf)
  • R. C. Kirby, A. Logg, L. R. Scott, and A. Terrel, “Topological optimization of the evaluation of finite element matrices,” SIAM J. Scientific Computing 28:224-240 (2006). (pdf)
  • R. C. Kirby, “Optimizing FIAT with Level 3 BLAS,” ACM Trans. Math. Software. 32:223–235 (2006). (pdf)
  • R. C. Kirby, M. G. Knepley, A. Logg, and L. R. Scott, “Optimizing the evaluation of finite element matrices,” SIAM J. Scientific Computing 27:741-758 (2005). (pdf)
  • R. C. Kirby, “FIAT: A new paradigm for computing finite element basis functions,” ACM Trans. Math. Software. 30:502-516 (2004). (pdf)
  • R. C. Kirby, “A new look at expression templates for matrix computation,” IEEE Computing in Science and Engineering, 5:66-70 (2003). (pdf)
  • R. Kirby, “A posteriori error estimates for the mixed finite element method,” Computational Geosciences. 7:197-214 (2003). (pdf)
  • R. Kirby, “On the convergence of high resolution methods with multiple time scales for hyperbolic conservation laws”, Math. Comp. 72:1239-1250 (2003). (pdf)
  • C. Dawson and R. Kirby, “High resolution schemes for conservation laws with locally varying time steps”, SIAM J. Sci. Comput. 22:2256-2281 (2001). (pdf)
  • C. Dawson and R. Kirby, “Solution of parabolic equations by backward-Euler mixed finite elements on a dynamically changing mesh”, SIAM J. Numer. Anal. 37:423-442 (2000). (pdf)
  • C. Dawson, S. Bryant, and R. Kirby, “Dynamically adaptive upwind finite volume methods for contaminant transport,” Computational Methods in Water Resources XII, vol. 2, 641-648 (1998). (pdf)

Book Chapters

I have contributed nine peer-reviewed chapters to the newly-released Springer book on the FEniCS project: Automated Solution of Differential Equations by the Finite Element Method (Logg, Mardal, Wells, eds).


Ph.D. students

  • Jorge Marchena-Menendez (defended March, 2023): From efficient high-order methods to scientific machine learning.  Now at Cambia Health Solutions.
  • Larry Allen (defended at Baylor, October 2021): Interpolation and approximation using Bernstein polynomials.  Now at NSA.
  • Tate Kernell (defended July 2019): Preconditioning mixed finite elements for tide models.
  • Brian Brennan (defended at Baylor, March 2015): Numerical analysis of a multi-physics model for trace gas sensors.  Now a Director of Data Science at Vinli.
  • Geoffrey Dillon (defended at TTU, June 2014): Schur complements and block preconditioners for coupled systems. Now at University of South Carolina.
  • Thinh Tri Kieu (defended at TTU, May 2014): Finite element methods for nonlinear wave equations. Now at University of Northern Georgia.

M.S. Students

Andy Terrel (joint with Ridg Scott, 2007), now Technical Executive at StoryFit