Volume 66, pp. 108-132, 2026.

Unstructured to structured: geometric multigrid on complex geometries via domain remapping

Nicolas Nytko, Scott MacLachlan, J. David Moulton, Luke N. Olson, Andrew Reisner, and Matthew West

Abstract

For domains that are easily represented by structured meshes, robust geometric multigrid solvers can quickly provide the numerical solution to many discretized elliptic PDEs. However, for complicated domains with unstructured meshes, constructing suitable hierarchies of meshes becomes challenging. We propose a framework for mapping computations from such complex domains to regular computational domains via diffeomorphisms, enabling the use of robust geometric-style multigrid. This mapping facilitates regular memory accesses during solves, improving efficiency and scalability, especially on massively parallel processors such as GPUs. Moreover, we show that the diffeomorphic mapping itself may be approximately learned using an invertible neural network, facilitating automated application to geometries where no analytic mapping is readily available.

Full Text (PDF) [1.3 MB], BibTeX , DOI: 10.1553/etna_vol66s108

Key words

geometric multigrid, diffeomorphic, LDDMM, neural ODE, structured grid

AMS subject classifications

65L60, 65N55