Date of Award
2026-05-01
Degree Name
Master of Science
Department
Computational Science
Advisor(s)
Peimeng Yin
Abstract
We investigate numerical methods for solving two-dimensional advection--diffusion--reaction equations using the Dynamical Low-Rank Finite Element Method (DLR-FEM) and the Dynamical Low-Rank Mass-Lumped Finite Element Method (DLR-MLFEM). The proposed framework combines dynamical low-rank approximation with finite element spatial discretization based on P_k elements, including linear P_1, quadratic P_2, and cubic P_3 basis functions, together with time integrators of varying order. While the DLR-FEM employs the standard consistent mass matrix, the DLR-MLFEM replaces it with a diagonal mass-lumped matrix and is advanced using a robust BUG-based integrator that exploits the simplified structure efficiently.
By restricting the numerical solution to a low-rank manifold of rank r, the proposed method significantly reduces the computational complexity from the full-rank formulation to O((m+n)r^2)), where m and n denote the degrees of freedom in the x- and y- directions, respectively. The well-posedness of the method is established, and an efficient fast solver is designed for the resulting low-rank systems. The methods are formulated and tested under homogeneous Neumann, homogeneous Dirichlet, and periodic boundary conditions, allowing a comprehensive assessment across different physical settings.
Numerical experiments with manufactured solutions confirm the expected convergence behavior in the L^2 -norm, namely O(Delta t + h^{k+1}), corresponding to second-, third-, and fourth-order spatial accuracy for P_1, P_2, and P_3 elements, respectively. In addition, the low-rank mass-lumped solver achieves substantial computational savings, with numerical results showing CPU-time reductions of approximately 95 % while maintaining accuracy comparable to full-rank methods. These results demonstrate that the combination of mass-lumped finite elements and dynamical low-rank approximation provides an accurate, stable, and scalable framework for large-scale advection--diffusion--reaction simulations.
Language
en
Provenance
Received from ProQuest
Copyright Date
2026-05
File Size
110 p.
File Format
application/pdf
Rights Holder
Joy Saha
Recommended Citation
Saha, Joy, "A Dynamical Low-Rank Mass-Lumped Finite Element Method For Advection-Diffusion-Reaction Equations" (2026). Open Access Theses & Dissertations. 4783.
https://scholarworks.utep.edu/open_etd/4783