Parallelization of improved variable-reduction method using GPU

In this study, we compare the parallel performance of the Improved Variable-Reduction Method (iVRM) and a variable preconditioned Krylov subspace method (VP Krylov) on a single GPU.We treat Poisson equations discretized by the element-free Galerkin method as target problems. iVRM avoids QR decomposition and eliminates Lagrange multipliers, enabling efficient solving of saddle-point systems dominated by sparse matrix operations. Numerical experiments show that both methods converge quickly, with iVRM exhibiting higher CPU efficiency. However, GPU results suggest that larger-scale problems are needed to fully utilize GPU resources. These findings guide the parallel design of iVRM for large-scale saddle-point problems.