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GATIPOR Report Summary

Project ID: 647134
Funded under: H2020-EU.1.1.

Periodic Reporting for period 1 - GATIPOR (Guaranteed fully adaptive algorithms with tailored inexact solvers for complex porous media flows)

Reporting period: 2015-09-01 to 2017-02-28

Summary of the context and overall objectives of the project

Efficient use of computational resources with a reliable outcome is a definite target in numerical simulations of partial differential equations (PDEs). Although this has been an important subject of numerical analysis and scientific computing for decades, still, surprisingly, often more than 90% of the CPU time in numerical simulations is literally wasted and the accuracy of the final outcome is not guaranteed. The reason is that addressing this complex issue rigorously is extremely challenging, as it stems from linking several rather disconnected domains like modeling, analysis of PDEs, numerical analysis, numerical linear algebra, and scientific computing. The goal of this project is to design novel inexact algebraic and linearization solvers, with each step being adaptively steered by optimal (guaranteed and robust) a posteriori error estimates, thus online interconnecting all parts of the numerical simulation of complex environmental porous media flows. The key novel ingredients will be multilevel algebraic solvers, tailored to porous media simulations, with problem- and discretization-dependent restriction, prolongation, and smoothing, yielding mass balance on all grid levels, accompanied by local adaptive stopping criteria. We shall theoretically prove the convergence of the new algorithms and justify their optimality, with in particular guaranteed (without any unknown constant) error reduction and overall computational load. Implementation into established numerical simulation codes and assessment on renowned academic and industrial benchmarks will consolidate the theoretical results. As a final outcome, the total simulation error will be certified and current computational burden cut by orders of magnitude. This would represent a cardinal technological advance both theoretically as well as practically in urgent environmental applications, namely the nuclear waste storage and the geological sequestration of CO2.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

"Several different issues have been addressed so far in the project:

a) We have first been focusing on estimating and localizing the algebraic component of the total error in numerical approximations. Our approach via algebraic error flux reconstructions has led to novel guaranteed upper and lower bounds on all the total error, as well as on its algebraic and discretization components. Novel ""safe"" stopping criteria for iterative solvers, ensuring that the algebraic error will lie below the dicretization one, were then conceived. These results are summarized in the preprint HAL 01312430 with J. Papež and Z. Strakoš, and in a paper to be submitted with J. Papež, U. Rüde, and B. Wohlmuth.

b) We have next focused on spatial hp-refinement strategies relying on the flux reconstructions methodology. In particular, strategies for the discontinuous Galerkin method were developed in a paper published in SIAM Journal on Scientific Computing 2016 38 (5), A3220–A3246 with V. Dolejší and A. Ern. Pursuing this topic in the framework of the Ph.D. thesis of P. Daniel and together with A. Ern and I. Smears, we can now also guarantee that the error will be reduced at least by a computable guaranteed factor on the next hp-step (paper to be submitted).

c) One of our central topics during this first period was the revisiting of a posteriori error analysis of time-dependent problems, namely of the model heat equation. We have in particular achieved the proof of efficiency for unsteady problems which is for the first time local with respect to both space and time, as intended in the project proposal. This forms the contents of two preprints (HAL 01377086 and 01489721), both with A. Ern and I. Smears.

d) I have also worked hard on the question of finding guaranteed a posteriori bounds for Laplace eigenvalues and eigenvectors. This has been achieved, in a robust way with respect to the polynomial degree, in the paper accepted for publication in SIAM Journal on Numerical Analysis in the conforming finite element setting and in a preprint (HAL 01483461) for basically any numerical method. These both results were obtained in collaboration with E. Cancès, G. Dusson, Y. Maday, and B. Stamm.

e) Three fundamental results underlying all of the above were also obtained: 1) stable broken H1 and H(div) polynomial extensions were obtained in the preprint HAL 01422204 with A. Ern; 2) a simple proof of localization of the W^{-1,q} norm with applications to inexact solvers was obtained in the preprint HAL 01332481 with J. Blechta and J. Málek; 3) discrete p-robust H(div)-liftings were constructed in the paper published recently in Calcolo, DOI 10.1007/s10092-017-0217-4, with A. Ern and I. Smears.

f) Finally, adaptive inexact iterative algorithms were proposed in the context of the Stokes problem in the preprint HAL 01097662, with M. Čermák, F. Hecht, and Z. Tang.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

a) To my knowledge, our results on algebraic error flux reconstructions are the first to allow for mass conservation at any step of any iterative linear algebraic solver. This means that even if an algebraic solver is stoped before reaching convergence, the output can be made completely physically meaningful, with in particular no mass lost. This is of special interest in the targetted porous media applications.

b) Deriving bounds on how much the error will (at least) be reduced in the next step is the key for proving convergence and optimality of adaptive numerical methods. To the best of our knowledge, the result in the paper in preparation is the first ever where such a bound is computable and guaranteed. Numerically, its precision turns out to be very high (overestimation by a factor very close to the optimal value of one).

c) Efficiency local with respect to both space and time has never been shown before. We believe that this opens a completely new door to analysis of adaptive numerical methods for time-dependent diffusion problems.

d) Eigenvalue problems are central in numerous applications but, so far, in contrast to source (elliptic boundary value) problems, no complete theory has been presented to obtain guaranteed a posteriori bounds for both eigenvalues and eigenvectors. This has now been achieved.

e) The main impacts of these three fundamental results are: 1) the broken polynomial extensions lead to the key polynomial-degree-robust efficiency in a posteriori error analysis of nonconforming methods and in three space dimensions; 2) localization of the a posteriori prediction of the error is now fully theoretically supported; 3) two particular consequences of this result are that the so-called transition condition that links two consecutive meshes in parabolic problems can now be removed and that meshes with arbitrary numbers of hanging nodes between elements can now be treated in a robust way (independent of the number of these hanging nodes).

f) Numerous iterations can be sparred by the developed inexact iterative algorithms, leading to important reductions of the current computational burden.

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