An hp Multigrid Approach for Tensor-Product Space-Time Finite Element Discretizations of the Stokes Equations

TL;DR

This paper introduces an innovative hp multigrid method for tensor-product space-time finite element discretizations of the Stokes equations, achieving high efficiency and scalability on large-scale problems.

Contribution

It develops a novel hp multigrid approach for space-time discretizations of the Stokes equations, utilizing matrix-free techniques and specialized smoothers for the first time.

Findings

Efficiently solves problems with over a trillion degrees of freedom.

Achieves high throughput on high-performance computing platforms.

Demonstrates robustness and scalability of the hp multigrid method.

Abstract

We present a monolithic $hp$ space-time multigrid method for tensor-product space-time finite element discretizations of the Stokes equations. Geometric and polynomial coarsening of the space-time mesh is performed, and the entire algorithm is expressed through rigorous mathematical mappings. For the discretization, we use inf-sup stable pairs $\mathbb Q_{r+1}/\mathbb P_{r}^{\text{disc}}$ of elements in space and a discontinuous Galerkin (DG$(k)$) discretization in time with piecewise polynomials of order $k$. The key novelty of this work is the application of $hp$ multigrid techniques in space and time, facilitated and accelerated by the matrix-free capabilities of the deal$.$II library. While multigrid methods are well-established for stationary problems, their application in space-time formulations encounter unique challenges, particularly in constructing suitable smoothers. To overcome these challenges, we employ space-time cell and vertex star patch based Vanka smoothers. Extensive tests on high-performance computing platforms demonstrate the efficiency of our \( hp \) multigrid approach on problem sizes exceeding a trillion degrees of freedom (dofs), sustaining throughputs of hundreds of millions of dofs per second.