Iterative Linear Solvers on GPUs

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Iterative Linear Solvers on GPUs#

Natalia Tylek — TU Wien

NGSolve User Meeting 2026, Winterthur

Note

This tutorial documents the CUDA-specific implementation as of June 2026.

Since then, the work has been folded into NGSolve’s general GPU layer. The ngsolve.gpu module now dispatches across backends — CUDA via ngscuda, Apple GPUs via ngsmetal, and a host reference device otherwise — and the graph-recording idea developed here has moved into that shared framework. The API may differ from what you see below.

For the current interface, see the official tutorial: Solving the Poisson Equation on devices

The material here remains useful for the reasoning behind the design: why per-iteration CPU synchronisation dominates GPU Krylov solvers, and how capturing the iteration loop removes it.

This presentation: two model problems demonstrating the use of CUDA Graph solvers

Problem

Solver

Status

Part 1

Poisson (symmetric SPD)

DevCGSolver

available in pre-release

Part 2

Convection (non-symmetric)

DevTFQMRSolver

(not) available in pre-release

Thesis context: both solvers are needed in the IPCS Navier–Stokes timestepper

NS component

Symmetric

Solver

GPU solver

Convection

No

TFQMR

DevTFQMRSolver

Viscous / mass

Yes (SPD)

CG + preconditioner

DevCGSolver

Pressure proj.

Yes (SPD)

CG + preconditioner

DevCGSolver

GPU Implementations and CUDA Graph Acceleration of Krylov Solvers for Incompressible Navier–Stokes in NGSolve

How to use#

  • Google Colab — no installation, runs in the browser
    Open in Colab

  • Local GPU — pip install ngsolve, then open the tutorial notebook

  • HPC cluster — see Getting Started for build and job submission

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