Space-time DPG methods for partial-differential equations with geophysical applications

Project Information

GEODPG

Grant agreement ID: 101017984

DOI

10.3030/101017984

Project closed

EC signature date 13 March 2021

Start date 1 January 2022

End date 3 June 2025

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost

€ 263 732,16

EU contribution

€ 263 732,16

Coordinated by

BCAM - BASQUE CENTER FOR APPLIED MATHEMATICS
Spain

Periodic Reporting for period 2 - GEODPG (Space-time DPG methods for partial-differential equations with geophysical applications)

Reporting period: 2024-01-01 to 2024-12-31

Summary of the context and overall objectives of the project

The crucial goal of the European commission’s climate action is to fight climate change. European Union’s target is to progressively reduce greenhouse emissions setting a cut goal of 55% by 2030 and 90% by 2040 compared to 1990 levels. Carbon dioxide (CO2) capture and sequestration is a long-term storage process that EU’s climate action supports and points out to be the only option to reduce unavoidable emissions in some industrial processes. In this process, it is critical to obtain a precise map of the Earth’s subsurface to characterize the possible sequestration sites (like depleted oil and gas reservoirs) in order to determine their condition and decide if they are suitable for storage. In most imaging techniques the acquired measurements are processed via numerical inversion to generate a subsurface map. One essential step in this process is the simulation of the forward problem governed by Partial Differential Equations (PDEs) via numerical methods. In this project, we seek to improve the characterization of Earth’s subsurface and its application to CO2 sequestration techniques by designing fast, stable, and accurate numerical methods to solve wave propagation problems.

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

Classical numerical methods for PDEs, like the Finite Element Method, may lead to instabilities in certain challenging problems (e.g. in advection-dominated-diffusion and wave propagation problems), and as a result, we might obtain non-physical solutions. In GEODPG, we focused on designing new computationally efficient and unconditionally stable numerical simulation solvers for approximating time-dependent PDEs. In particular, we developed a class of methods based on the Discontinuos Petrov-Galerkin (DPG) methodology. A robust derivation and an efficient implementation of the DPG method as a time-marching scheme was previously unexplored. During the first two years of the project, we successfully developed efficient and stable solvers for both parabolic and hyperbolic problems, together with adaptive strategies to accurately approximate predefined quantities of interest. Moreover, we developed routines for approximating exponential-related functions of matrices (a key step in our simulations) that are orders of magnitude faster than the current state-of-the-art.

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)

We have met all the goals set in the project for the first two years. We have developed new numerical methods for simulating time-dependent PDEs with excellent approximation and stability properties. We have also developed adaptive strategies to accurately approximate physically relevant features of the solution of the PDEs and reduce the error in some user-defined quantity of interest. For the last year of the project, we will employ the developed solvers to solve wave propagation models in geophysics. The final goal of GEODPG is to develop better techniques to characterize the Earth’s subsurface for geophysical applications including CO2-sequestration. Therefore, the impact of the intersectoral and interdisciplinary collaborations within the fellowship will contribute to the 13th sustainable development goal of United Nations about the action to combat climate change. Applications that can also benefit from a precise characterization of the subsurface include earthquake prediction and seismic hazard estimation, mine detection and geothermal energy production.

Wave propagation problem solved with the DPG-based time-marching scheme