Peter Bastian

Dominic Kempf, René Heß, Steffen Müthing et al.

SIMD vectorization has lately become a key challenge in high-performance computing. However, hand-written explicitly vectorized code often poses a threat to the software's sustainability. In this publication we solve this sustainability and performance portability issue by enriching the simulation framework dune-pdelab with a code generation approach. The approach is based on the well-known domain-specific language UFL, but combines it with loopy, a more powerful intermediate representation for the computational kernel. Given this flexible tool, we present and implement a new class of vectorization strategies for the assembly of Discontinuous Galerkin methods on hexahedral meshes exploiting the finite element's tensor product structure. The optimal variant from this class is chosen by the code generator through an autotuning approach. The implementation is done within the open source PDE software framework Dune and the discretization module dune-pdelab. The strength of the proposed approach is illustrated with performance measurements for DG schemes for a scalar diffusion reaction equation and the Stokes equation. In our measurements, we utilize both the AVX2 and the AVX512 instruction set, achieving 40\% to 60\% of the machine's theoretical peak performance for one matrix-free application of the operator.

Lukas Holbach, Peter Bastian, Robert Scheichl

We introduce a two-level hybrid restricted additive Schwarz (RAS) preconditioner for heterogeneous steady-state convection-diffusion equations at high Péclet numbers. Our construction builds on the multiscale spectral generalized finite element method (MS-GFEM), wherein the coarse space is spanned by locally optimal basis functions obtained from local generalized eigenproblems on operator-harmonic spaces. Extending the theory of Ma (2025) to convection-diffusion problems in conservation form, we establish exponential convergence of the MS-GFEM approximation with respect to the dimension of the local approximation space. Rewriting MS-GFEM as a RAS-type iteration, we show for coercive problems that this exponential convergence property is inherited by the RAS-type iterative method (at least in the continuous setting). Employed as a preconditioner within the generalized minimal residual method (GMRES), the resulting method requires only a few iterations for high accuracy even with low-dimensional coarse spaces. Through extensive numerical experiments on problems with high-contrast diffusion and non-divergence-free, rotating velocity fields, we demonstrate robustness with respect to the grid Péclet number and the number of subdomains (tested up to $10^5$ subdomains), while coarse-space dimensions remain small as grid Péclet numbers increase. By adapting the coarse space and oversampling size, we are able to achieve arbitrarily fast convergence of preconditioned GMRES. As an extension, for which we do not have theory yet, we show effectiveness of the method even for indefinite problems and in the vanishing-diffusion limit.

Marian Piatkowski, Steffen Müthing, Peter Bastian

In this paper we consider discontinuous Galerkin (DG) methods for the incompressible Navier-Stokes equations in the framework of projection methods. In particular we employ symmetric interior penalty DG methods within the second-order rotational incremental pressure correction scheme. The major focus of the paper is threefold: i) We propose a modified upwind scheme based on the Vijayasundaram numerical flux that has favourable properties in the context of DG. ii) We present a novel postprocessing technique in the Helmholtz projection step based on $H(\text{div})$ reconstruction of the pressure correction that is computed locally, is a projection in the discrete setting and ensures that the projected velocity satisfies the discrete continuity equation exactly. As a consequence it also provides local mass conservation of the projected velocity. iii) Numerical results demonstrate the properties of the scheme for different polynomial degrees applied to two-dimensional problems with known solution as well as large-scale three-dimensional problems. In particular we address second-order convergence in time of the splitting scheme as well as its long-time stability.