1) A posteriori estimates on the error in numerical approximation with quality uniform in the polynomial degree
I was working extensively on computable estimates that are robust in the polynomial degree p of the numerical approximation. We have obtained such estimates in new settings including the H1 and H(div) spaces (SIAM Journal on Scientific Computing 38 (2016), A3220–A3246; Calcolo 54 (2017), 1009-1025; Mathematics of Computation 89 (2020), 551-594; IMA Journal of Numerical Analysis (2021), DOI 10.1093/imanum/draa103) and H(curl) spaces (Comptes Rendus Mathématique. Académie des Sciences. Paris 358 (2020) 1101-1110; Mathematics of Computation 91 (2022), 37-74; Calcolo 58 (2021), 53; HAL Preprint 03227570 (2021)), as well as for the Helmholtz problem (Numerische Mathematik 148 (2021), 525-573), the advection (ESAIM. Mathematical Modelling and Numerical Analysis 55 (2021), S447-S474) and diffusion-reaction problems (ESAIM. Mathematical Modelling and Numerical Analysis 54 (2020), 1951-1973), the eigenvalue problems (SIAM Journal on Numerical Analysis 55 (2017), 2228-2254; Numerische Mathematik 140 (2018), 1033-1079; Mathematics of Computation 89 (2020), 2563-2611), and the parabolic heat problem (SIAM Journal on Numerical Analysis 55 (2017), 2811-2834; IMA Journal of Numerical Analysis 39 (2019), 1158-1179). With growing computer power, where increased polynomial degrees can be easily employed, this shall be the methodology for the future.
2) Polynomial-degree-robust and adaptively steered multilevel algebraic solvers
We have also introduced our ideas on error control and its use to steer the algorithm into numerical linear algebra solvers. The solvers in SIAM Journal on Numerical Analysis 58 (2020), 2856-2884; SIAM Journal on Scientific Computing 43 (2021), S117-S145; and Computational Methods in Applied Mathematics 21 (2021), 445-468 inherently provide reliable and efficient a posteriori error estimates of the algebraic error, which in turn adaptively guides the solver behavior. Moreover, they are p-robust. I again believe that this shall be the methodology for the future.
3) Robust a posteriori control for inexact solvers
We were able to fully theoretically support the localization of our a posteriori prediction of the error for model nonlinear problems, in a way robust with respect to the nonlinearity. Moreover, inexact algebraic solvers are covered. These results from IMA Journal of Numerical Analysis 40 (2020), 914-950 & ESAIM. Mathematical Modelling and Numerical Analysis 52 (2018), 2037-2064 certify that the adaptive mesh refinement will concentrate the effort to the right place.
4) A posteriori error estimates distinguishing the spatial, temporal, regularization, linearization, and algebraic components and fully adaptive algorithms
Following the project proposal, we were able to certify the total error committed in the computer simulation on any step of the algorithm and to steer it adaptively in order to reduce the usual computational burden. The desired estimates and algorithms were derived in the context of linear elliptic problems (Numerische Mathematik 138 (2018), 681-721; Computational Methods in Applied Mathematics 18 (2018), 495-519; Computer Methods in Applied Mechanics and Engineering 371 (2020), 113243; Journal of Computational and Applied Mathematics 366 (2020), 112367), the Stokes problem (Numerische Mathematik 138 (2018), 1027-1065), degenerate parabolic problems in the form of a nonlinear anisotropic Fokker–Planck equation (Mathematics of Computation 90 (2021), 517-563), doubly-degenerate Richards equation (HAL Preprint 03328944 (2021)), and problems involving steady and unsteady variational inequalities (Journal of Scientific Computing 84 (2020), 28; Computer Methods in Applied Mechanics and Engineering 367 (2020), 113105).
5) A posteriori error analysis for time-dependent problems
We have achieved the proof of efficiency for unsteady problems which is local with respect to both time and space (SIAM Journal on Numerical Analysis 55 (2017), 2811-2834 and IMA Journal of Numerical Analysis 39 (2019), 1158-1179), as well applications to space-time domain decomposition methods in Electronic Transactions on Numerical Analysis 49 (2018), 151-181 and HAL Preprint 03355088.
6) Convergence and optimality with adaptive inexact solvers
For a model nonlinear elliptic problem, we in Numerische Mathematik 147 (2021), 679-725 prove optimal decay of the numerical approximation error with respect to the overall cost invested into the computer simulation. We have also managed to formulate an (hp) adaptive algorithm with guaranteed convergence (guaranteed error reduction) (Computers & Mathematics with Applications 76 (2018), 967-983; Computer Methods in Applied Mechanics and Engineering 359 (2020), 112607; HAL Preprint 02486433).
7) Applications to highly complex situations related to urgent environmental problems
Again along the lines of the proposal, I have finally focused on applications of the developed methodology to challenging real-life environmental problems (SMAI Journal of Computational Mathematics 5 (2019), 195-227; Computer Methods in Applied Mechanics and Engineering 331 (2018), 728-760; Computational Geosciences 24 (2020), 1031-1055; HAL Preprint 03355116).
Exploitation and dissemination of the achieved results
All the achieved results are published or submitted for publication in highly-valued international journals and presented at international conferences. I believe we have obtained a nice number of fundamental results for difficult problems that were completely open a few years ago. The algorithms based on error control and adaptive steering of all error components were immediately implemented in academic scientific computing codes, as well as into production codes used by our industrial partners at ANDRA and IFPEN for an immediate use in practice.
