Preconditioning Techniques for Hybridizable Discontinuous Galerkin Discretizations on GPU Architectures

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.