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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - GATIPOR (Guaranteed fully adaptive algorithms with tailored inexact solvers for complex porous media flows)

Teaser

Summary

Introduction

This project studies algorithms for numerical approximation of complex systems of unsteady nonlinear partial differential equations, typically arising in underground porous media problems. It aims at designing novel inexact algebraic and linearization solvers, with each step being adaptively steered, interconnecting at any time all parts of the numerical simulation algorithm. Its key ingredient are optimal a posteriori estimates on the error in the approximate solution which give a guaranteed global upper bound, guaranteed local lower bounds, robustness with respect to the problem parameters, and which distinguish the different error components like the spatial, temporal, regularization, linearization, and algebraic solver ones.

Tools

* novel multilevel algebraic solvers tailored to porous media simulations
* novel adaptive inexact Newton linearizations
* mass balance on each algorithm step
* guaranteed and robust a posteriori error estimates
* full adaptivity with local stopping criteria
* theoretical proofs of convergence and optimality
* guaranteed error reduction and optimal overall computational load
* interconnection of modeling, analysis of PDEs, numerical analysis, numerical linear algebra, and scientific computing

Final goals

The first final goal is to certify the total error on any step of a numerical simulation. The second final goal is to reduce in an important way the current computational burden.

Impact

On the theoretical side, the results impact the overall perception of numerical simulations of model partial differential equations: here efficient use of computational resources with a reliable outcome is a definite target. 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. On the practical side, we consider applications to important contemporary environmental problems like underground storage of dangerous waste or geological sequestration of CO2. Here we showcase computer implementation and assess the developed methodology on academic and industrial benchmarks, often in collaboration with industrial partners. Indeed, the total simulation error can be certified even for these complex problems, and the current computational burden can be cut by orders of magnitude.

Work performed

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

a) We have first focused on consolidating the theory of a posteriori error estimates via potential and flux reconstructions, while analyzing inhomogeneous Dirichlet and Neumann boundary conditions, varying polynomial degree, and mixed rectangular-triangular grids possibly containing hanging nodes. Asymptotic exactness is observed for smooth solutions and uniform mesh refinement, whereas optimal exponential convergence rates are reported for singular solutions and adaptive hp-refinement. In particular, strategies for the discontinuous Galerkin method were developed in a paper published in SIAM Journal on Scientific Computing 38 (2016), A3220â€“A3246 with V. DolejÅ¡Ã 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 (Computers & Mathematics with Applications 76 (2018), 967-983).

b) We have next focused on estimating and localizing the total error and its algebraic and discretization components. Guaranteed upper and lower bounds and novel \"\"safe\"\" stopping criteria for iterative solvers were conceived. Polynomial-degree robustness is established. These results are summarized in the paper with J. PapeÅ¾ and Z. StrakoÅ¡, Numerische Mathematik 138 (2018), 681-721, and HAL Preprint 01662944 with J. PapeÅ¾, U. RÃ¼de, and B. Wohlmuth. Similar results for the Stokes problem have been obtained in Numerische Mathematik 138 (2018), 1027-1065, together with M. ÄŒermÃ¡k, F. Hecht, and Z. Tang. In a series of papers (Computational Methods in Applied Mathematics 18 (2018), 495-519, with S. Ali Hassan, C. Japhet, and M. Kern; Electronic Transactions on Numerical Analysis 49 (2018), 151-181, with S. Ali Hassan and C. Japhet; and HAL Preprint 01540956, with E. Ahmed, S. Ali Hassan, C. Japhet, and M. Kern), we then apply this methodology to domain decomposition methods. These results are particularly tailored to Robin and Ventcell transmission conditions and to global-in-time formulations that allow for local time stepping and parallelization in time. Even for degenerate parabolic problem with nonlinear and discontinuous transmission conditions, guaranteed a posteriori estimates valid at each iteration and distinguishing the different error components are obtained.

c) Two fundamental results underlying some of the above developments were also obtained: 1) stable broken H1 and H(div) polynomial extensions, in the preprint HAL 01422204 with A. Ern; 2) discrete p-robust H(div)-liftings, in the paper in Calcolo 54 (2017), 1009-1025 (post-doc of I. Smears and with A. Ern).

d) 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 time and space, as intended in the project proposal. This forms the contents of the paper in SIAM Journal on Numerical Analysis 55 (2017), 2811-2834, and IMA Journal of Numerical Analysis (2018), DOI 10.1093/imanum/dry035, both following the post-doc of I. Smears and with A. Ern.

e) Dual norms like the dual norm of the residual and the distance norm to the H^1_0 space seem to be fundamentally global over the entire computational domain. In M2AN. Mathematical Modelling and Numerical Analysis (2018), DOI 10.1051/m2an/2018034, together with P. Ciarlet, we prove, though, that they are both equivalent to the Hilbertian sums of their localizations over patches of elements. In HAL Preprint 01332481 with J. Blechta and J. MÃ¡lek, we extend this result from the Hilbert space H^1_0 to the Sobolev space W^{1,p}_0, with p bigger than or equal to one, and to an arbitrary bounded linear functional on W^{1,p}_0.

f) In accordance with the pro\"

Final results

a) Deriving bounds on how much the error will (at least) be reduced in the next step of an adaptive loop is the key for proving convergence and optimality of adaptive numerical methods. To the best of our knowledge, the result in Computers & Mathematics with Applications 76 (2018), 967-983, obtained in the framework of the Ph.D. thesis of Patrik Daniel and in collaboration with A. Ern and I. Smears, 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).

b) To my knowledge, our results on total and algebraic error estimates are the first to 1) give guaranteed error bounds; 2) be polynomial-degree rebust; 3) ensure that the algebraic error will lie below the dicretization one; 4) yield mass conservation on any step of any iterative linear algebraic solver, so 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 (of special interest in the targeted porous media applications). The outcome of the simulation is consequently certified at any point, and important savings of iterations are realized.

c) The discrete p-robust H(div)-liftings results allow to remove the so-called transition condition linking two consecutive meshes in a posteriori analysis of parabolic problems. For elliptic problems, they allow for robust treatment of arbitrary number of hanging nodes. These results are completely new to the best of our knowledge. The constructive proof is based on a posteriori analysis of an auxiliary elliptic problem with a H^-1 source term, thereby yielding results of independent interest.

d) Efficiency local with respect to both time and space 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.

e) Localization of the a posteriori prediction of the error is now fully theoretically supported, even in presence of inexact algebraic solvers. This entails robustness of a posteriori estimates for noncoercive and nonlinear partial differential equations in divergence form.

f) Our developments for locally conservative methods on polytopal meshes lead to an easy-to-implement and fast-to-run adaptive algorithm with guaranteed overall precision, adaptive stopping criteria thanks to distinction of error components, and adaptive space and time mesh refinements, in an industrial context of the targeted porous media applications.

g) The developed adaptive inexact semismooth Newton method numerically leads to tight overall error control and important computational savings. We believe that it sets a paradigm on how non-differentiable algebraic system arising from complementarity constraints should be treated in an optimal way.

h) 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. A particular impact of these results lies in computational chemistry first-principle molecular simulations, where they allow to bound the ground-state energy in a sharp and certified way. Extensions to multiple eigenvalues and clusters, as well as to more complex nonlinear eigenvalue problems, are in progress.