Unlike the traditional finite element method, which relies on an explicitly given approximation space (typically piecewise polynomial), in multiscale numerical methods the approximation space is driven numerically by the PDE model, incorporating fine-scale details of the domain geometry and coefficient distribution. The multiscale methods developed by the associate team can be interpreted as approximate substructuring techniques, where the interiors of macro-cells are eliminated through a low-dimensional parametrization of either Neumann data (as in MHM [1,3]) or Dirichlet data (as in Trefftz methods [2]). Since the computation of the approximation basis is local to the coarse cells, multiscale numerical methods are highly parallelizable, which allows them to benefit from increasing computational facilities while keeping communications very low. Alternatively, multiscale basis functions can be “learned” using machine learning techniques, which makes multiscale methods even more accessible.
The research program of this postdoctoral position will focus on error analysis of multiscale numerical discretization methods for nonlinear problems, and their integration with domain decomposition and scientific machine learning approaches. In terms of applications, the research program aims to go beyond simplified academic examples and address problems such as subsurface multiphase flow and contact mechanics.
[1] Araya, R., Harder, C., Paredes, D., & Valentin, F. (2013). Multiscale hybrid-mixed method. SIAM Journal on Numerical Analysis, 51(6), 3505-3531.
[2] Boutilier, M., Brenner, K., & Dolean, V. (2024). Robust methods for multiscale coarse approximations of diffusion models in perforated domains. Applied Numerical Mathematics, 201, 561-578.
[3] Gomes, A. T. A., Pereira, W. S., & Valentin, F. (2023). The MHM method for linear elasticity on polytopal meshes. IMA Journal of Numerical Analysis, 43(4), 2265-2298.
