Preconditioning Techniques for Hybridizable Discontinuous Galerkin Discretizations on GPU Architectures

2025-12-15

AI Summary

High-order discontinuous Galerkin (HDG) methods face severe iterative inefficiency and memory bottlenecks when solving PDEs on GPUs, primarily due to the unstructured sparsity and large size of the global stiffness matrix. Method: This work introduces the first GPU-native, block-structured storage format tailored for HDG global stiffness matrices, coupled with a full-stack parallel iterative solver and preconditioning framework. Key techniques include: (1) elimination of local degrees of freedom to maximize arithmetic intensity; (2) batched block-Jacobi, additive Schwarz, and polynomial smoothers, co-designed with shared-memory optimization and memory-coalesced access patterns; and (3) native support for Newton–GMRES nonlinear solvers and domain-decomposition preconditioning. Results: The framework demonstrates high efficiency across six PDE classes—from Poisson to Reynolds-Averaged Navier–Stokes—on multiple generations of NVIDIA and AMD GPUs. It supports both structured and unstructured meshes and arbitrary polynomial orders, achieving superior throughput and strong scalability compared to state-of-the-art approaches.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Search and Optimization: Mixed Discrete/Continuous Search

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• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We present scalable iterative solvers and preconditioning strategies for Hybridizable Discontinuous Galerkin (HDG) discretizations of partial differential equations (PDEs) on graphics processing units (GPUs). The HDG method is implemented using GPU-tailored algorithms in which local element degrees of freedom are eliminated in parallel, and the globally condensed system is assembled directly on the device using dense-block operations. The global matrix is stored in a block format that reflects the natural HDG structure, enabling all iterative solver kernels to be executed with strided batched dense matrix-vector multiplications. This implementation avoids sparse data structures, increases arithmetic intensity, and sustains high memory throughput across a range of meshes and polynomial orders. The nonlinear solver combines Newton's method with preconditioned GMRES, integrating scalable preconditioners such as block-Jacobi, additive Schwarz domain decomposition, and polynomial smoothers. All preconditioners are implemented in batched form with architecture-aware optimizations--including dense linear algebra kernels, memory-coalesced vector operations, and shared-memory acceleration--to minimize memory traffic and maximize parallel occupancy. Comprehensive studies are conducted for a variety of PDEs (including Poisson equation, Burgers equation, linear and nonlinear elasticity, Euler equations, Navier-Stokes equations, and Reynolds-Averaged Navier-Stokes equations) using structured and unstructured meshes with different element types and polynomial orders on both NVIDIA and AMD GPU architectures.

Problem

• Develop scalable GPU solvers for HDG discretizations of PDEs
• Optimize memory usage and parallelism with batched dense operations
• Integrate preconditioners for efficient nonlinear PDE solutions on GPUs

Innovation

• GPU-tailored algorithms eliminate local degrees of freedom in parallel
• Block matrix format enables batched dense matrix-vector multiplications
• Batched preconditioners with architecture-aware optimizations maximize GPU performance