High-performance conjugate gradient solver with two-level multiplicative preconditioner for finite element problems

We present a preconditioned conjugate gradient solver with a two-level multiplicative preconditioner for symmetric positive-definite systems arising from finite element discretizations. The method combines a block-Jacobi smoother at the fine level with a smoothed aggregation algebraic multigrid coarse correction. A mixed-precision strategy uses single-precision arithmetic in the preconditioner while maintaining double-precision CG iterations. All solver parameters are determined automatically from the matrix structure, with no problem-specific tuning. Numerical experiments on real-world and synthetic FEM benchmarks demonstrate competitive wall-clock performance relative to a leading commercial solver and robustness across various block sizes, strong anisotropy, and large condition numbers.