Oxford FEM reading group schedule and suggested texts


See NA arXiv for today.

Kirby, R.C., Klöckner, A. and Sepanski, B., 2021. Finite elements for Helmholtz equations with a nonlocal boundary condition. SIAM Journal on Scientific Computing, 43(3), pp.A1671–A1691.

Bacuta, C., 2006. A unified approach for Uzawa algorithms. SIAM Journal on Numerical Analysis, 44(6), pp.2633–2649.

Guzmán, J. and Neilan, M., 2014. Conforming and divergence-free Stokes elements on general triangular meshes. Mathematics of Computation, 83(285), pp.15–36.

Holst, M., Maxwell, D., and Mazzeo, R., 2017. Conformal fields and the structure of the space of solutions of the Einstein constraint equations. arXiv preprint arXiv:1711.01042.

Angoshtari, A. and Yavari, A., 2016. Hilbert complexes of nonlinear elasticity. Zeitschrift für angewandte Mathematik und Physik, 67(6), pp.1–30.

Arnold, D. N. and Brezzi, F., 1985. Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. ESAIM: Mathematical Modelling and Numerical Analysis, 19(1), pp.7–32,
or the more modern/unified work: Cockburn, B., Gopalakrishnan, J., and Lazarov, R., 2009. Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems. SIAM Journal on Numerical Analysis, 47(2), pp.1319–1365.

Beirão da Veiga, L., Dassi, F., Manzini, G., and Mascotto, L., 2022. The Virtual Element Method for the 3D Resistive Magnetohydrodynamic Model. arXiv preprint arXiv:2201.04417.

Bewley, T., Mashayek, A., Cavaglieri, D., and Luchini, P., 2021. Tweed and wireframe: accelerated relaxation algorithms for multigrid solution of elliptic PDEs on stretched structured grids. arXiv preprint arXiv:2110.08389.

Neunteufel, M., Schöberl, J., and Sturm, K., 2021. Numerical shape optimization of the Canham–Helfrich–Evans bending energy. arXiv preprint arXiv:2107.13794.

Liu, H., Neilan, M., and Otus, B., 2022. A divergence-free finite element method for the Stokes problem with boundary correction. Journal of Numerical Mathematics.

Kean, K., Neilan, M., and Schneier, M., 2021. The Scott–Vogelius Method for the Stokes Problem on Anisotropic Meshes. arXiv preprint arXiv:2109.14780.

Arnold, D. N., Brezzi, F., Cockburn, B., and Marini, L. D., 2002. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis, 39(5), pp.1749–1779.

Arnold, D., Falk, R., and Winther, R., 1997. Preconditioning in H(div) and applications. Mathematics of Computation, 66(219), pp.957–984.

Neilan, M. and Wu, M., 2019. Discrete Miranda–Talenti estimates and applications to linear and nonlinear PDEs. Journal of Computational and Applied Mathematics, 356, pp.358–376.

Xu, J., 2001. The method of subspace corrections. Journal of Computational and Applied Mathematics, 128(1-2), pp.335–362.

Hu, X., Xu, J., and Zikatanov, L.T., 2019. Randomized and fault-tolerant method of subspace corrections. Research in the Mathematical Sciences, 6(3), pp.1–15.

Morgan, H. and Scott, L. R., 2018. Towards a unified finite element method for the Stokes equations. SIAM Journal on Scientific Computing, 40(1), pp.A130–A141.

Amrouche, C., Ciarlet, P. G., and Mardare, C., 2015. On a lemma of Jacques-Louis Lions and its relation to other fundamental results. Journal de Mathématiques Pures et Appliquées, 104(2), pp.207–226.

Ortner, C. and Süli, E., 2007. Discontinuous Galerkin finite element approximation of nonlinear second-order elliptic and hyperbolic systems. SIAM Journal on Numerical Analysis, 45(4), pp.1370–1397.

Holst, M. and Stern, A., 2012. Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces. Foundations of Computational Mathematics, 12(3), pp.263–293.
Sequel paper: Holst, M. and Stern, A., 2012. Semilinear mixed problems on Hilbert complexes and their numerical approximation. Foundations of Computational Mathematics, 12(3), pp.363–387.

Hughes, T. J., Cottrell, J. A., and Bazilevs, Y., 2005. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194(39-41), pp.4135–4195.

Buffa, A., Dölz, J., Kurz, S., Schöps, S., Vázquez, R., and Wolf, F., 2020. Multipatch approximation of the de Rham sequence and its traces in isogeometric analysis. Numerische Mathematik, 144(1), pp.201–236.

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.

Carstensen, C., 2006. Clément interpolation and its role in adaptive finite element error control. Partial Differential Equations and Functional Analysis (pp. 27–43). Birkhäuser Basel.

Allen, L. and Kirby, R. C., 2021. Bounds-constrained polynomial approximation using the Bernstein basis. arXiv preprint arXiv:2104.11819.

Cockburn, B. and Xia, S., 2021. An adjoint-based super-convergent Galerkin approximation of eigenvalues. Journal of Computational Physics, 449, article 110816.

[SF, (21/2 or 7/3)/23]

[AB-R, (14 or 21)/2/23]

[UZ, (7 or 14)/2/23]

[IM, (31/1 or 7/2)/23]

[CPII, (24 or 31)/1/23]

[FA, (17 or 24)/1/23]

[BA, (10 or 17)/1/23]

[AF, 13/12/22]

[PBM, 6/12/22]

[PF, 29/11/22]

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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:

Kay, D., 1997. The p-and hp-finite element method applied to a class of non-linear elliptic partial differential equations. (Doctoral thesis, University of Leicester).

Any chapter of
Li, L., 2018. Regge finite elements with applications in solid mechanics and relativity (Doctoral dissertation, University of Minnesota).

Ch. 6 onwards of
Boffi, D., Brezzi, F., and Fortin, M., 2013. Mixed Finite Element Methods and Applications (Vol. 44, pp. xiv–685). Heidelberg: Springer.

Ch. 5 onwards of
Elman, H. C., Silvester, D. J., and Wathen, A. J., 2014. Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Oxford Science Publications.

Ch. 8–13 (mesh generation and interpolation) of
Ern, A. and Guermond, J. L., 2021. Finite elements I: Approximation and interpolation (Vol. 72). Springer Nature.

Part X (eigenproblems) of
Ern, A. and Guermond, J. L., 2021. Finite Elements II: Galerkin approximation, elliptic and mixed PDEs (Vol. 73). Springer Nature.

Part XII (Friedrichs systems; stabilisations) of
Ern, A. and Guermond, J. L., 2021. Finite Elements III: First-Order and Time-Dependent PDEs (Vol. 74). Springer Nature.

[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.