Oxford FEM reading group schedule and suggested texts

Papers:

Schedule:

✓ [IP, 8/8/23] Qin, O. and Xu, K., 2023. Solving nonlinear ODEs with the ultraspherical spectral method. arXiv preprint arXiv:2306.17688.

✓ [BA, 1/8/23] Gawlik, E. S. and Gay-Balmaz, F., 2021. A variational finite element discretization of compressible flow. Foundations of Computational Mathematics, 21, pp.961–1001.

✓ [DS, 18/7/23] Fu, G., Guzmán, J., and Neilan, M., 2020. Exact smooth piecewise polynomial sequences on Alfeld splits. Mathematics of Computation, 89(323), pp.1059–1091.

✓ [PAG-O, 11/7/23] Barrenechea, G., Georgoulis, E., Pryer, T., and Veeser, A., 2023. A nodally bound-preserving finite element method. arXiv preprint arXiv:2304.01067.

✓ [IM, 4/7/23] Rognes, M. E., Ham, D. A., Cotter, C. J., and McRae, A. T., 2013. Automating the solution of PDEs on the sphere and other manifolds in FEniCS 1.2. Geoscientific Model Development, 6(6), pp.2099–2119.

✓ [AB-R, 20/6/23] Bernardi, C., Laval, F., Métivet, B., and Pernaud-Thomas, B., 1992. Finite element approximation of viscous flows with varying density. SIAM Journal on Numerical Analysis, 29(5), pp.1203–1243.

✓ [AF, 13/6/23] Bacuta, C., 2006. A unified approach for Uzawa algorithms. SIAM Journal on Numerical Analysis, 44(6), pp.2633–2649.

✓ [AN, 6/6/23] Ni, A., 2022. Fast adjoint differentiation of chaos via computing unstable perturbations of transfer operators. arXiv preprint arXiv:2111.07692.

✓ [PBM, 30/5/23] Farrell, P. E., 2017. Mathematical background: adjoints and their applications, dolfin-adjoint documentation.

✓ [UZ, 16/5/23] Gopalakrishnan, J., Lederer, P. L., and Schöberl, J., 2020. A mass conserving mixed stress formulation for Stokes flow with weakly imposed stress symmetry. SIAM Journal on Numerical Analysis, 58(1), pp.706–732.

✓ [FA, 9/5/23] Stoter, S.K., Cockburn, B., Hughes, T.J., and Schillinger, D., 2022. Discontinuous Galerkin methods through the lens of variational multiscale analysis. Computer Methods in Applied Mechanics and Engineering, 388, p.114220.

✓ [CPII, 2/5/23] Ainsworth, M., Andriamaro, G., and Davydov, O., 2011. Bernstein–Bézier finite elements of arbitrary order and optimal assembly procedures. SIAM Journal on Scientific Computing, 33(6), pp.3087–3109.

✓ [IP, 25/4/23] Burns, K. J., Vasil, G. M., Oishi, J. S., Lecoanet, D., and Brown, B.P., 2020. Dedalus: A flexible framework for numerical simulations with spectral methods. Physical Review Research, 2(2), p.023068.

✓ [PBM, 11/4/23] Budisa, A., Hu, X., Kuchta, M., Mardal, K.-A., and Zikatanov, L., 2022. Rational approximation preconditioners for multiphysics problems. arXiv preprint arXiv:2209.11659.
using also: Holter, K. E., Kuchta, M., and Mardal, K.-A., 2020. Robust preconditioning of monolithically coupled multiphysics problems. arXiv preprint arXiv:2001.05527.

✓ [KH, 28/3/23] Talk on distributional elements, using Pechstein, A. S. and Schöberl, J., 2018. An analysis of the TDNNS method using natural norms. Numerische Mathematik, 139, pp.93–120.

✓ [PAG-O, 21/3/23] Ang, A., De Sterck, H. and Vavasis, S., 2023. MGProx: A nonsmooth multigrid proximal gradient method with adaptive restriction for strongly convex optimization. arXiv preprint arXiv:2302.04077.

✓ [DS, 14/3/23] Borden, M. J., Scott, M. A., Evans, J. A., and Hughes, T. J., 2011. Isogeometric finite element data structures based on Bézier extraction of NURBS. International Journal for Numerical Methods in Engineering, 87(1–5), pp.15–47.

✓ [AB-R, 7/3/23] Pollock, S., Rebholz, L. G., and Xiao, M., 2019. Anderson-accelerated convergence of Picard iterations for incompressible Navier–Stokes equations. SIAM Journal on Numerical Analysis, 57(2), pp.615–637.

✓ [UZ, 21/2/23] Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L. D., and Russo, A., 2013. Basic principles of virtual element methods. Mathematical Models and Methods in Applied Sciences, 23(01), pp.199–214.

✓ [IM, 14/2/23] Crum, J., Cheng, C., Ham, D. A., Mitchell, L., Kirby, R.C., Levine, J. A., and Gillette, A., 2022. Bringing trimmed serendipity methods to computational practice in Firedrake. ACM Transactions on Mathematical Software (TOMS), 48(1), pp.1–19,
using also: Gillette, A., Kloefkorn, T., and Sanders, V., 2019. Computational serendipity and tensor product finite element differential forms. The SMAI Journal of Computational Mathematics, 5, pp.1–21.

✓ [AF, 7/2/23] Neunteufel, M., Schöberl, J., and Sturm, K., 2021. Numerical shape optimization of the Canham–Helfrich–Evans bending energy. arXiv preprint arXiv:2107.13794.

✓ [FA, 31/1/23] Plaza, A. and Carey, G.F., 2000. Local refinement of simplicial grids based on the skeleton. Applied Numerical Mathematics, 32(2), pp.195–218.

✓ [BA, 24/1/23] La Scala, G., 2022. Variational time-steppers that are finite element in time for Firedrake and Irksome. (Masters dissertation, Imperial College London).

✓ [CPII, 17/1/23] Babuška, I. and Suri, M., 1994. The p and h-p versions of the finite element method, basic principles and properties. SIAM Review, 36(4), pp.578–632.

✓ [PBM, 6/12/22] Brubeck, P.D. and Farrell, P.E., 2022. Multigrid solvers for the de Rham complex with optimal complexity in polynomial degree. arXiv preprint arXiv:2211.14284.

✓ [PF, 29/11/22] Spillane, N., Dolean, V., Hauret, P., Nataf, F., Pechstein, C. and Scheichl, R., 2014. Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps. Numerische Mathematik, 126(4), pp.741–770,
using also: Butler, R., Dodwell, T., Reinarz, A., Sandhu, A., Scheichl, R. and Seelinger, L., 2020. High-performance dune modules for solving large-scale, strongly anisotropic elliptic problems with applications to aerospace composites. Computer Physics Communications, 249, p.106997.

✓ [KH, 22/11/22] Bounded Poincaré operators for BGG complexes (joint work with Andreas Čap).

✓ [IP, 15/11/22] Bourdin, B., 2001. Filters in topology optimization. International Journal for Numerical Methods in Engineering, 50(9), pp.2143–2158.
Summary: One of the classical problems in topology optimization is the minimization of the compliance of a linearly elastic material. The model is ill-posed in general and needs to be coupled with a "restriction method" to ensure the existence of solutions. A popular one is the use of a density filter. Bourdin proved that density filters guarantee the existence of a minimizer and also showed some basic FEM convergence results :)

✓ [PAG-O, 8/11/22] Bartels, S. and Wang, Z., 2021. Orthogonality relations of Crouzeix–Raviart and Raviart–Thomas finite element spaces. Numerische Mathematik, 148(1), pp.127–139,
using also: Bartels, S., 2021. Nonconforming discretizations of convex minimization problems and precise relations to mixed methods. Computers & Mathematics with Applications, 93, pp.214–229.

✓ [AB-R, 1/11/22] Baier-Reinio, A., Rhebergen, S. and Wells, G. N., 2022. Analysis of pressure-robust embedded-hybridized discontinuous Galerkin methods for the Stokes problem under minimal regularity. Journal of Scientific Computing, 92(2), pp.1–25.

✓ [UZ, 25/10/22] Boffi, D., Brezzi, F. and Gastaldi, L., 1997. On the convergence of eigenvalues for mixed formulations. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 25(1-2), pp.131–154.
GitHub repo of slides and numerical examples.

✓ [CPII, 18/10/22] Ainsworth, M. and Parker, C., 2022. Unlocking the secrets of locking: Finite element analysis in planar linear elasticity. Computer Methods in Applied Mechanics and Engineering, 395, p.115034.

✓ [FA, 11/10/22] Arnold, D. N. and Walker, S. W., 2020. The Hellan–Herrmann–Johnson method with curved elements. SIAM Journal on Numerical Analysis, 58(5), pp.2829–2855.

✓ [CPII, 4/10/22] Ainsworth, M. and Parker, C., 2021. Mass conserving mixed hp-FEM approximations to Stokes flow. Part I: Uniform stability. SIAM Journal on Numerical Analysis, 59(3), pp.1218–1244.

✓ [IP, 23/8/22] Bertrand, F. and Boffi, D., 2022. On the necessity of the inf-sup condition for a mixed finite element formulation. arXiv preprint arXiv:2206.06968.

✓ [KH, 16/8/22] Talk on tensor product finite element BGG complexes.

✓ [PF, 9/8/22] Hiptmair, R. and Xu, J., 2007. Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. SIAM Journal on Numerical Analysis, 45(6), pp.2483–2509.

✓ [PBM, 5/7/22] Moler, C. and Van Loan, C., 2003. Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review, 45(1), pp.3–49,
and: Min, M. and Fischer, P., 2013. An efficient high-order time integration method for spectral-element discontinuous Galerkin simulations in electromagnetics. Journal of Scientific Computing, 57(3), pp.582–603.

✓ [PAG-O, 28/6/22] John, V., Li, X., Merdon, C., and Rui, H., 2022. Inf-sup stabilized Scott–Vogelius pairs on general simplicial grids by Raviart–Thomas enrichment. arXiv preprint arXiv:2206.01242,
using as introduction: Li, X. and Rui, H., 2021. A low-order divergence-free H(div)-conforming finite element method for Stokes flows. IMA Journal of Numerical Analysis, 00, pp1–24.

✓ [FA, 21/6/22] Ch. 9: Trees and directed acyclic graphs, of Ham, D. A., 2021. Object-oriented programming in python for mathematicians. Independently published.

✓ [IP, 14/6/22] Di Nezza, E., Palatucci, G., and Valdinoci, E., 2012. Hitchhikerʼs guide to the fractional Sobolev spaces. Bulletin des Sciences Mathématiques, 136(5), pp.521–573.

✓ [PBM, 7/6/22] Ainsworth, M. and Jiang, S., 2021. Preconditioning the mass matrix for high order finite element approximation on tetrahedra. SIAM Journal on Scientific Computing, 43(1), pp.A384–A414.

✓ [FL, 31/5/22] Margenberg, N., Hartmann, D., Lessig, C., and Richter, T., 2022. A neural network multigrid solver for the Navier–Stokes equations. Journal of Computational Physics, p.110983.

✓ [BA, 24/5/22] Bonizzoni, F. and Kanschat, G., 2021. H¹-conforming finite element cochain complexes and commuting quasi-interpolation operators on Cartesian meshes. Calcolo, 58(2), pp.1–29.

✓ [PAG-O, 17/5/22] Ern, A. and Vohralík, M., 2013. Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs. SIAM Journal on Scientific Computing, 35(4), pp.A1761–A1791.

✓ [GGDD, 10/5/22; sessions now hybrid] Continuing with: Bui-Thanh, T., Ghattas, O., Martin, J., and Stadler, G., 2013. A computational framework for infinite-dimensional Bayesian inverse problems part I: The linearized case, with application to global seismic inversion. SIAM Journal on Scientific Computing, 35(6), pp.A2494–A2523.

✓ [KH, 3/5/22] De Rham complexes of Lie-group-invariant differential forms (work in progress with Y.-J. Lee).

✓ [FA, 26/4/22] Hiptmair, R., Pauly, D., and Schulz, E., 2022. Traces for Hilbert complexes. arXiv preprint arXiv:2203.00630.

✓ [IP, 5/4/22] Olver, S. and Townsend, A., 2013. A fast and well-conditioned spectral method. SIAM Review, 55(3), pp.462–489.

✓ [PF, 29/3/22] Dobrev, V., Kolev, T., Lee, C. S., Tomov, V., and Vassilevski, P. S., 2019. Algebraic hybridization and static condensation with application to scalable H(div) preconditioning. SIAM Journal on Scientific Computing, 41(3), pp.B425–B447.

✓ [PBM, 22/3/22] Pazner, W., 2020. Efficient low-order refined preconditioners for high-order matrix-free continuous and discontinuous Galerkin methods. SIAM Journal on Scientific Computing, 42(5), pp.A3055–A3083.

✓ [FL, 15/3/22] Longo, M., Opschoor, J. A., Disch, N., Schwab, C., and Zech, J., 2022. De Rham compatible deep neural networks. arXiv preprint arXiv:2201.05395,
using as introduction: He, J., Li, L., Xu, J. and Zheng, C., 2020. ReLU deep neural networks and linear finite elements. Journal of Computational Mathematics, 38(3), pp.502–527.

✓ [BA, 8/3/22] Arnold, D. N., Falk, R. S., and Winther, R., 2000. Multigrid in H(div) and H(curl). Numerische Mathematik, 85(2), pp.197–217.
Summary: The essential premise is deriving certain inequalities and bounds so as to prove that, so long as appropriate FE spaces and smoothers are used, then multigrid gives an efficient solver/preconditioner for problems in H(div) & H(curl).

✓ [GGDD, 1/3/22] Bui-Thanh, T., 2012. A gentle tutorial on statistical inversion using the Bayesian paradigm. Institute for Computational Engineering and Sciences, Technical Report ICES-12-18,
discussing also: Bui-Thanh, T., Ghattas, O., Martin, J., and Stadler, G., 2013. A computational framework for infinite-dimensional Bayesian inverse problems part I: The linearized case, with application to global seismic inversion. SIAM Journal on Scientific Computing, 35(6), pp.A2494–A2523.

✓ [PAG-O, 22/2/22] Walkington, N. J., 2010. Compactness properties of the DG and CG time stepping schemes for parabolic equations. SIAM Journal on Numerical Analysis, 47(6), pp.4680–4710,
using also: Part XIII, Ch. 69–71 (DG/Continuous PG in time; inf-sup analysis) of Ern, A. and Guermond, J. L., 2020. Finite elements III: First-order and time-dependent PDEs (Vol. 74). Springer Nature.

✓ [FA, 15/2/22] Neunteufel, M. and Schöberl, J., 2021. Avoiding membrane locking with Regge interpolation. Computer Methods in Applied Mechanics and Engineering, 373, p.113524,
using also: Ch. 8 of Neunteufel, M., 2021. Mixed finite element methods for nonlinear continuum mechanics and shells (Doctoral dissertation, Institute of Technical Mechanics, Johannes Kepler University Linz Wien).

✓ [KH, 8/2/22] Talk concerning geometry and complexes, starting with: Arnold, D. N., 2002. A quick introduction to the Einstein equations. Lecture notes for the IMA Workshop on General Relativity, June 2002.

✓ [PBM, 1/2/22] Moura, R. C., Cassinelli, A., da Silva, A. F., Burman, E., and Sherwin, S. J., 2022. Gradient jump penalty stabilisation of spectral/hp element discretisation for under-resolved turbulence simulations. Computer Methods in Applied Mechanics and Engineering, 388, p.114200.

✓ [IP, 25/1/22] Kreuzer, C. and Georgoulis, E., 2018. Convergence of adaptive discontinuous Galerkin methods. Mathematics of Computation, 87(314), pp.2611–2640,
and some of: Kawecki, E.L. and Smears, I., 2021. Convergence of adaptive discontinuous Galerkin and C⁰-interior penalty finite element methods for Hamilton–Jacobi–Bellman and Isaacs equations. Foundations of Computational Mathematics, pp.1–50.

✓ [FL, 18/1/2022] Gleason, T. A., Peters, E. L., and Evans, J.A., 2022. A divergence-conforming hybridized discontinuous Galerkin method for the incompressible magnetohydrodynamics equations. arXiv preprint arXiv:2201.01906,
using as introduction: Gibson, T. H., Mitchell, L., Ham, D. A., and Cotter, C. J., 2020. Slate: extending Firedrake's domain-specific abstraction to hybridized solvers for geoscience and beyond. Geoscientific Model Development, 13(2), pp.735–761.

✓ [PH, 7/12/21] Wang, J. and Zhang, R., 2012. Maximum principles for P1-conforming finite element approximations of quasi-linear second order elliptic equations. SIAM Journal on Numerical Analysis, 50(2), pp.626–642.

✓ [BA, 30/11/21] Champneys, A. R. and Sandstede, B., 2007. Numerical computation of coherent structures. In Numerical continuation methods for dynamical systems (pp. 331–358). Springer, Dordrecht.

✓ [GGDD, 23/11/21] Gustafsson, T., Stenberg, R., and Videman, J., 2017. On finite element formulations for the obstacle problem – Mixed and stabilised methods. Computational Methods in Applied Mathematics, 17(3), pp.413–429.
Companion paper to: Gustafsson, T., Stenberg, R., and Videman, J., 2017. Mixed and stabilized finite element methods for the obstacle problem. SIAM Journal on Numerical Analysis, 55(6), pp.2718–2744.
Summary: The obstacle problem can be solved with penalty methods, a Nitsche type method, or via mixed methods via a Lagrange multiplier (with the possibility of stabilising the FE formulation). There seem to be many connections with other classical methods, like imposing Dirichlet BCs via a Lagrange multiplier or via Nitsche's method.

✓ [FA, 16/11/21] First half of Cockburn, B., 1998. An introduction to the discontinuous Galerkin method for convection-dominated problems. In Advanced numerical approximation of nonlinear hyperbolic equations, (pp. 150–268). Springer, Berlin, Heidelberg.

✓ [PBM, 9/11/21] Auricchio, F., Da Veiga, L. B., Lovadina, C., Reali, A., Taylor, R. L., and Wriggers, P., 2013. Approximation of incompressible large deformation elastic problems: some unresolved issues. Computational Mechanics, 52(5), pp.1153–1167.

✓ [IP, 2/11/21] Buffa, A. and Ortner, C., 2009. Compact embeddings of broken Sobolev spaces and applications. IMA Journal of Numerical Analysis, 29(4), pp.827–855.
Summary: Suppose you want to discretise a PDE-constrained optimisation problem with FEM, where the minimisers live in H¹, and you want to prove your discretisation converges to the minimiser. If you have a H¹-conforming discretisation, somewhere along the way you will probably have to extract an L²-strongly converging sequence from a H¹ weakly converging sequence using Rellich–Kondrachov. However, if you are not using a H¹-conforming discretisation but you have interior penalisation (like Brezzi–Douglas–Marini or Raviart–Thomas for Stokes), this trick is no longer available. Turns out you can still extract an L²-strongly converging sequence anyway.

✓ [FL, 26/10/21] Mitusch, S. K., Funke, S. W., and Kuchta, M, 2021. Hybrid FEM-NN models: Combining artificial neural networks with the finite element method. Journal of Computational Physics, 446, p.110651.
Summary: [IP] Discretise spatial dimensions with FEM and use neural networks to estimate unknown coefficients/parameters, e.g. heat diffusion coefficient. Seems to work better than using PINNs directly and potentially better than a pointwise optimisation approach. Current open problems are quadrature, preconditioning, & an implementation in PyTorch/Tensorflow.

✓ [GGDD, 19/10/21] Linke, A., Merdon, C., and Neilan, M., 2019. Pressure-robustness in quasi-optimal a priori estimates for the Stokes problem. arXiv preprint arXiv:1906.03009.
Summary: [IP] If you are not using a divergence-free discretisation for Stokes, then you should be using reconstruction operators. Change my mind!

✓ [FA, 12/10/21] Yamazaki, H., Weller, H., Cotter, C. J., and Browne, P. A., 2021. Conservation with moving meshes over orography. arXiv preprint arXiv:2108.00805.
Summary: For finite volume discretisations of conservation laws over topographic surfaces, the practice of r-adapting a mesh (via optimal transportation theory) can violate preservation of the volume of the domain and total mass, but this can be remedied by solving for a 'volume adjustment parameter', and transport equations for each cell volume.

✓ [KH, 25/06/2021] FEEC-related topics, including: Arnold, D.N. and Hu, K., 2021. Complexes from complexes. Foundations of Computational Mathematics, pp.1–36.

Book/thesis chapters:

✓ [02/2021 – 06/2021] Bartels, S., 2015. Numerical methods for nonlinear partial differential equations (Vol. 47). Berlin: Springer.

✓ [10/2020 – 02/2021] Arnold, D. N., 2018. Finite element exterior calculus. Society for Industrial and Applied Mathematics.

✓ [24/20/2020] Video meeting with L.R.S.
✓ [01/2020 – 06/2020] Brenner, S. C., and Scott, L.R., 2008. The mathematical theory of finite element methods (Vol. 3). New York: Springer.