Paid Internship F/H: High-Order Hybrid Continuous/Discontinuous Galerkin Spectral Finite Element Methods for Seismic Wave Simulation

The Makutu project team specializes in large-scale simulations applied to the reconstruction of
complex media, used to gain a better understanding of the internal dynamics of environments that are
difficult or even impossible to probe. To this end, it develops advanced numerical methods that are
integrated into open-source platforms deployed on state-of-the-art HPC environments.

This internship aims to develop and validate a high-order hybrid continuous/discontinuous Galerkin spectral finite element solver for seismic wave simulation, with applications in subsurface imaging and CO₂ monitoring. The project combines the efficiency of continuous Galerkin methods and the robustness of discontinuous Galerkin approaches, targeting improved accuracy, reduced computational cost, and enhanced scalability on modern HPC architectures. Key deliverables include a validated prototype solver, benchmark reports, and performance studies, while participants will gain expertise in advanced numerical methods, seismic workflows, and high-performance computing.

Seismic imaging is a critical technology for subsurface characterization in exploration geophysics, CO₂ storage monitoring, and underground hazard assessment. The industry increasingly relies on full waveform inversion (FWI) and reverse time migration (RTM), which require repeated and highly accurate solutions of the acoustic (and elastic) wave equation on large-scale 2D/3D domains.

Finite difference solvers dominate current industrial codes, but their limitations are well known:

• Poor performance on complex geometries and topography.
• High numerical dispersion for high-frequency modeling.
• Difficulty in coupling with adaptive meshes.

High-order spectral finite element methods (SFEM) offer superior accuracy per degree of freedom and are naturally suited to HPC architectures (CPU/GPU clusters). Two main Galerkin formulations exist:

• Continuous Galerkin (CG-SFEM): Memory-efficient and fast in homogeneous regions.
• Discontinuous Galerkin (DG-SFEM): Robust near discontinuities, complex geology, and allows local adaptivity, but at higher computational cost.

A hybrid CG/DG approach could combine the efficiency of CG with the robustness of DG, yielding a competitive industrial solver that reduces time-to-solution while maintaining seismic accuracy requirements.

This internship is a great opportunity to develop the following skills:

• Advanced knowledge of numerical methods for wave propagation.
• Experience with high-order Galerkin spectral methods (CG and DG).
• Hands-on HPC programming (MPI, OpenMP, CUDA).
• Understanding of seismic workflows (modeling, FWI, RTM).

This will be facilitated by collaborating with members of Makutu team.

The expected deliverables are : 

• Prototype hybrid solver (validated, scalable, with documented source code).
• Benchmark report comparing CG, DG, and hybrid CG/DG on seismic test cases.
• HPC scalability study with recommendations for evolution.
• Final internship report & presentation with emphasis on industrial impact.

The internship will deliver a proof-of-concept hybrid CG/DG spectral finite element solver for the acoustic wave equation. The developments will extend the SFEM wave equation platform based on CG SFEM  formulation to DG SFEM formulation. The different tasks will be:

• Prototype implementation of a hybrid CG/DG-SFEM solver targeting 3D seismic wave propagation in complex media
• Quantitative evaluation of accuracy, dispersion, and stability versus existing solvers.
• HPC performance benchmarking (MPI, GPU acceleration) with focus on scalability to large meshes.
• Integration scenarios for seismic imaging workflows (forward modeling, FWI/RTM kernels).
• Mathematical & Numerical Formulation: Develop the variational form of the acoustic wave equation; Implement DG-SFEM solvers (high-order Lagrange basis with Gauss–Lobatto quadrature); Design CG/DG interface treatment.
• Prototype Development: Build the solver in the wave equation platform developed by Makutu. This platform is developed in C++ and uses Kokkos to expose the computational kernels to GPUS
• Benchmarking and Validation: Validate against reference solutions; Test seismic benchmarks; Compare with pure CG SFEM implementation.
• Application Testing : Apply solver to realistic seismic modeling scenario; Evaluate impact on imaging kernels (RTM/FWI forward modeling); Document trade-offs between accuracy and performance.

The expected impact is:

• Reduced computational cost: Hybrid approach may lower CPU/GPU requirements for large-scale seismic modeling.
• Improved imaging accuracy: High-order spectral methods reduce numerical dispersion, crucial for high-frequency content in FWI.
• HPC readiness: The solver design will be optimized for modern heterogeneous clusters (CPU/GPU).
• Innovation opportunity: Potential integration into next-generation industrial seismic imaging platforms.

 Key references:

• Komatitsch, D., & Tromp, J. (2002). Spectral-element simulations of global seismic wave propagation—I. Validation. J. Int.
• De Basabe, J. D., & Sen, M. K. (2007). Grid dispersion and stability criteria of some common finite‐element methods for acoustic and elastic wave equations. Geophysics.
• Chaljub, E., et al. (2007). Spectral element analysis in seismology. Advances in Geophysics.
• Warburton, T., & Hesthaven, J. S. (2008). Nodal discontinuous Galerkin methods: algorithms, analysis, and applications. Springer.
• Abdi, R., et al. (2021). GPU-accelerated spectral-element method for seismic wave propagation. Computers & Geosciences.

Required profile
This internship is intended for Master’s students (typically at the second-year level, M2 or equivalent) with solid technical training in numerical methods and advanced scientific computing.

Languages
Working proficiency in French and/or English.

Soft skills
We are looking for a motivated candidate who enjoys teamwork, values knowledge sharing, and is eager to contribute to a collaborative research environment.

Additional assets
The ability to clearly communicate results, both through oral presentations and written reports, will be highly appreciated.

• Subsidized meals
• Partial reimbursement of public transport costs
• Possibility of teleworking and flexible organization of working hours
• Professional equipment available (videoconferencing, loan of computer equipment, etc.)
• Social, cultural and sports events and activities

Depending on the amount of the gratuity in effect.