A stencil scaling approach for accelerating matrix-free finite element implementations

TL;DR

This paper introduces a new operator scaling method for matrix-free finite element assembly that significantly reduces computational cost while maintaining accuracy, demonstrated through large-scale high-performance computing applications.

Contribution

The paper presents a novel stencil scaling approach for finite element assembly that improves efficiency and preserves convergence properties in variable coefficient PDEs.

Findings

Requires about one third of the floating-point operations of classical methods.

Achieves asymptotically optimal convergence in $H^1$ and $L^2$ norms.

Successfully applied to large-scale problems with over 100 billion degrees of freedom.

Abstract

We present a novel approach to fast on-the-fly low order finite element assembly for scalar elliptic partial differential equations of Darcy type with variable coefficients optimized for matrix-free implementations. Our approach introduces a new operator that is obtained by appropriately scaling the reference stiffness matrix from the constant coefficient case. Assuming sufficient regularity, an a priori analysis shows that solutions obtained by this approach are unique and have asymptotically optimal order convergence in the $H^1$- and the $L^2$-norm on hierarchical hybrid grids. For the pre-asymptotic regime, we present a local modification that guarantees uniform ellipticity of the operator. Cost considerations show that our novel approach requires roughly one third of the floating-point operations compared to a classical finite element assembly scheme employing nodal integration. Our theoretical considerations are illustrated by numerical tests that confirm the expectations with respect to accuracy and run-time. A large scale application with more than a hundred billion ($1.6\cdot10^{11}$) degrees of freedom executed on 14,310 compute cores demonstrates the efficiency of the new scaling approach.