Low-cost non-iterative Sylvester solvers for dynamical low-rank approximation of reaction-diffusion equations

Abstract

In this work, we develop low-cost non-iterative Sylvester solvers for dynamical low-rank approximation of reaction-diffusion equations and incorporate them into a basis-update \& Galerkin mass-lumped finite element method (\mbox{BUG-MLFEM}). By incorporating mass lumping via Gauss-Lobatto quadrature, the spatial discretization yields a matrix differential equation equipped with diagonal positive-definite mass matrices. We establish the weighted dynamical low-rank approximation (WDLRA) within the Galerkin framework of the MLFEM system, proving the theoretical equivalence between the BUG integrator and MLFEM on low-rank subspaces, which guarantees the discrete well-posedness of the implicit BUG steps. To overcome the severe computational bottleneck of standard $O(m^3+n^3)$ Sylvester solvers, we develop tailored, low-cost non-iterative solvers that exploit the inherent sparse-by-small algebraic structure. Under Dirichlet and Neumann boundary conditions (BCs), the large coefficient matrices are banded, enabling efficient diagonalization of the small $r \times r$ factor matrix and reducing the per-step computational complexity to $O((m+n)r^2)$ and storage to $O((m+n)r)$. For periodic BCs, we combine this diagonalization strategy with a rank-2 Sherman-Morrison-Woodbury update to handle wrap-around corner entries at negligible additional cost. Numerical experiments demonstrate first-order temporal and $(k+1)$th-order spatial convergence of the proposed BUG-MLFEM in the discrete $L^2$ norm, where $k$ denotes the polynomial degree of the finite element space, together with significant computational speedups over existing low-rank integrators.