Two-level preconditioning with diffusion-filtered coarse spaces for ill-conditioned finite-element SPD systems

This work proposes an algebraic two-level preconditioning strategy for large sparse symmetric positive definite (SPD) linear systems arising in finite element simulations. The method constructs a reusable, problem-adaptive coarse space by applying a diffusion filter to a set of probe vectors, and then extracting a compact basis via truncated singular value decomposition (SVD). The diffusion filter acts as a spectral low-pass, yielding a subspace concentrated in slowly decaying, near-null components that typically govern the asymptotic convergence of Krylov methods. To couple this coarse space with a chosen baseline solver, we introduce two complementary construction strategies: Strategy A incorporates baseline information through a mixed set of raw and preconditioner-transformed probes, whereas Strategy B incorporates the baseline at the operator level by filtering the left-preconditioned operator. In both cases, the compressed basis is integrated with standard preconditioners through a balanced deflation/two-level operator. We evaluate the approach on a suite of benchmark matrices and report iteration counts and wall-clock times per solve with and without the proposed correction. The experiments demonstrate that the two-level layer can markedly reduce iterations on ill-conditioned problems, while highlighting practical trade-offs between coarse-space setup cost, per-iteration overhead, and robustness across matrix families, showcasing its suitability to amortized settings.