Non-invasive multigrid for semi-structured grids

TL;DR

This paper extends multigrid solvers to semi-structured meshes, enabling more efficient and flexible solutions for complex geometries while maintaining compatibility with existing structured multigrid methods.

Contribution

It introduces a formal mathematical framework for semi-structured multigrid, making it easier to adapt existing applications and demonstrating potential performance improvements.

Findings

The semi-structured multigrid closely relates to classical multigrid methods.

Numerical experiments show potential performance gains.

The framework clarifies software design for semi-structured solvers.

Abstract

Multigrid solvers for hierarchical hybrid grids (HHG) have been proposed to promote the efficient utilization of high performance computer architectures. These HHG meshes are constructed by uniformly refining a relatively coarse fully unstructured mesh. While HHG meshes provide some flexibility for unstructured applications, most multigrid calculations can be accomplished using efficient structured grid ideas and kernels. This paper focuses on generalizing the HHG idea so that it is applicable to a broader community of computational scientists, and so that it is easier for existing applications to leverage structured multigrid components. Specifically, we adapt the structured multigrid methodology to significantly more complex semi-structured meshes. Further, we illustrate how mature applications might adopt a semi-structured solver in a relatively non-invasive fashion. To do this, we propose a formal mathematical framework for describing the semi-structured solver. This formalism allows us to precisely define the associated multigrid method and to show its relationship to a more traditional multigrid solver. Additionally, the mathematical framework clarifies the associated software design and implementation. Numerical experiments highlight the relationship of the new solver with classical multigrid. We also demonstrate the generality and potential performance gains associated with this type of semi-structured multigrid.