Matrix-Free Evaluation Strategies for Continuous and Discontinuous Galerkin Discretizations on Unstructured Tetrahedral Grids

Abstract

This study presents novel strategies for improving the node-level performance of matrix-free evaluation of continuous and discontinuous Galerkin spatial discretizations on unstructured tetrahedral grids. In our approach the underlying integrals of a generic finite-element operator are computed cell-by-cell through numerical quadrature using tabulated dense local matrices of shape functions, achieving high throughput for low to moderate-order polynomial degrees. By employing dense matrix-matrix products instead of matrix-vector products for the cell-wise interpolation, the method reaches over $60\%$ of peak performance. The optimization strategies exploit explicit data parallelism to enhance computational efficiency, complemented by a hierarchical mesh reordering algorithm that improves data locality. The matrix-free implementation achieves up to a $6\times$ speedup compared to a global sparse matrix-based approach at a polynomial degree of three. The effectiveness of the method is demonstrated through numerical experiments on the Poisson and Navier--Stokes equations. The Poisson operator is preconditioned by a hybrid multigrid scheme that combines auxiliary continuous finite-element spaces, polynomial and geometric coarsening where possible while employing algebraic multigrid on the coarse mesh. Within the preconditioner, the implementation transitions between the matrix-free and matrix-based strategies for optimal efficiency. Finally, we analyze the strong scaling behavior of the Poisson and Helmholtz operators, demonstrating the method's potential to solve large real-world problems.