arXiv subjects

Seulip Lee

Publications and source records attributed to Seulip Lee.

8 recordsLinked to original sources

arXiv2026

Fully discrete analysis of pressure-robust enriched Galerkin methods for the time-dependent Stokes equations

This paper presents a fully discrete analysis of pressure-robust enriched Galerkin (EG) methods for the time-dependent Stokes equations. The spatial discretization employs a velocity reconstruction operator that preserves the continuous component of the EG space while mapping its enriched discontinuous component into the lowest-order Raviart-Thomas ($\mathcal{R}T_0$) subspace of $H(\mathrm{div})$, ensuring strict local mass conservation and pressure-robustness. Fully discrete schemes are constructed using the backward Euler and Crank-Nicolson methods, with the reconstruction incorporated into both the forcing and discrete time-derivative terms. We prove unconditional stability and derive optimal-order, parameter-explicit \textit{a priori} error estimates for both the velocity and pressure. In particular, the velocity estimates are independent of the continuous pressure and the irrotational component of the forcing term, contain no inverse-viscosity factors, and explicitly track the spatial mesh size, time-step size, and viscosity. Numerical experiments in two and three dimensions verify the predicted convergence rates, strict local mass conservation, and robustness with respect to the viscosity, while comparisons with non-pressure-robust variants demonstrate the importance of incorporating the reconstruction into the discrete time-derivative term.

math.NA↗

arXiv2025

A unified spatiotemporal formulation with physics-preserving structure for time-dependent convection-diffusion problems

We propose a unified four-dimensional (4D) spatiotemporal formulation for time-dependent convection-diffusion problems that preserves underlying physical structures. By treating time as an additional space-like coordinate, the evolution problem is reformulated as a stationary convection-diffusion equation on a 4D space-time domain. Using exterior calculus, we extend this framework to the full family of convection-diffusion problems posed on $H(\textbf{grad})$, $H(\textbf{curl})$, and $H(\text{div})$. The resulting formulation is based on a 4D Hodge-Laplacian operator with a spatiotemporal diffusion tensor and convection field, augmented by a small temporal perturbation to ensure nondegeneracy. This formulation naturally incorporates fundamental physical constraints, including divergence-free and curl-free conditions. We further introduce an exponentially-fitted 4D spatiotemporal flux operator that symmetrizes the convection-diffusion operator and enables a well-posed variational formulation. Finally, we prove that the temporally-perturbed formulation converges to the original time-dependent convection-diffusion model as the perturbation parameter tends to zero.

math.AP↗

arXiv2025

A monotone finite element method for an elliptic distributed optimal control problem with a convection-dominated state equation

We propose and analyze a monotone finite element method for an elliptic distributed optimal control problem constrained by a convection-diffusion-reaction equation in the convection-dominated regime. The method is based on the edge-averaged finite element (EAFE) scheme, which is known to preserve the discrete maximum principle for convection-diffusion problems. We show that the EAFE discretization inherits the monotonicity property of the continuous problem and consequently preserves the desired-state bounds at the discrete level, ensuring that the numerical optimal state remains stable and free of nonphysical oscillations. The discrete formulation is analyzed using a combination of the EAFE consistency result and a discrete inf-sup condition, which together guarantee well-posedness and yield the optimal convergence order. Comprehensive numerical experiments are presented to confirm the theoretical findings and to demonstrate the robustness of the proposed scheme in the convection-dominated regimes.

math.NA↗

arXiv2025

A $C^0$ weak Galerkin method with preconditioning for constrained optimal control problems with general tracking

This paper presents a $C^0$ weak Galerkin ($C^0$-WG) method combined with an additive Schwarz preconditioner for solving optimal control problems (OCPs) governed by partial differential equations with general tracking cost functionals and pointwise state constraints. These problems pose significant analytical and numerical challenges due to the presence of fourth-order variational inequalities and the reduced regularity of solutions. Our first contribution is the design of a $C^0$-WG method based on globally continuous quadratic Lagrange elements, enabling efficient elementwise stiffness matrix assembly and parameter-free implementation while maintaining accuracy, as supported by a rigorous error analysis. As a second contribution, we develop an additive Schwarz preconditioner tailored to the $C^0$-WG method to improve solver performance for the resulting ill-conditioned linear systems. Numerical experiments confirm the effectiveness and robustness of the proposed method and preconditioner for both biharmonic and optimal control problems.

math.NA↗

arXiv2024

Edge-averaged virtual element methods for convection-diffusion and convection-dominated problems

This manuscript develops edge-averaged virtual element (EAVE) methodologies to address convection-diffusion problems effectively in the convection-dominated regime. It introduces a variant of EAVE that ensures monotonicity (producing an $M$-matrix) on Voronoi polygonal meshes, provided their duals are Delaunay triangulations with acute angles. Furthermore, the study outlines a comprehensive framework for EAVE methodologies, introducing another variant that integrates with the stiffness matrix derived from the lowest-order virtual element method for the Poisson equation. Numerical experiments confirm the theoretical advantages of the monotonicity property and demonstrate an optimal convergence rate across various mesh configurations.

math.NA↗

arXiv2023

A uniform and pressure-robust enriched Galerkin method for the Brinkman equations

This paper presents a pressure-robust enriched Galerkin (EG) method for the Brinkman equations with minimal degrees of freedom based on EG velocity and pressure spaces. The velocity space consists of linear Lagrange polynomials enriched by a discontinuous, piecewise linear, and mean-zero vector function per element, while piecewise constant functions approximate the pressure. We derive, analyze, and compare two EG methods in this paper: standard and robust methods. The standard method requires a mesh size to be less than a viscous parameter to produce stable and accurate velocity solutions, which is impractical in the Darcy regime. Therefore, we propose the pressure-robust method by utilizing a velocity reconstruction operator and replacing EG velocity functions with a reconstructed velocity. The robust method yields error estimates independent of a pressure term and shows uniform performance from the Stokes to Darcy regimes, preserving minimal degrees of freedom. We prove well-posedness and error estimates for both the standard and robust EG methods. We finally confirm theoretical results through numerical experiments with two- and three-dimensional examples and compare the methods' performance to support the need for the robust method.

math.NA↗

arXiv2023

Pressure-robust enriched Galerkin methods for the Stokes equations

In this paper, we present a pressure-robust enriched Galerkin (EG) scheme for solving the Stokes equations, which is an enhanced version of the EG scheme for the Stokes problem proposed in [Son-Young Yi, Xiaozhe Hu, Sanghyun Lee, James H. Adler, An enriched Galerkin method for the Stokes equations, Computers and Mathematics with Applications, accepted, 2022]. The pressure-robustness is achieved by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. An a priori error analysis proves that the velocity error is independent of the pressure and viscosity. We also propose and analyze a perturbed version of our pressure-robust EG method that allows for the elimination of the degrees of freedom corresponding to the discontinuous component of the velocity vector via static condensation. The resulting method can be viewed as a stabilized $H^1$-conforming $\mathbb{P}_1$-$\mathbb{P}_0$ method. Further, we consider efficient block preconditioners whose performances are independent of the viscosity. The theoretical results are confirmed through various numerical experiments in two and three dimensions.

math.NA↗

arXiv2022

A low-cost, parameter-free, and pressure-robust enriched Galerkin method for the Stokes equations

In this paper, we propose a low-cost, parameter-free, and pressure-robust Stokes solver based on the enriched Galerkin (EG) method with a discontinuous velocity enrichment function. The EG method employs the interior penalty discontinuous Galerkin (IPDG) formulation to weakly impose the continuity of the velocity function. However, the symmetric IPDG formulation, despite of its advantage of symmetry, requires a lot of computational effort to choose an optimal penalty parameter and to compute different trace terms. In order to reduce such effort, we replace the derivatives of the velocity function with its weak derivatives computed by the geometric data of elements. Therefore, our modified EG (mEG) method is a parameter-free numerical scheme which has reduced computational complexity as well as optimal rates of convergence. Moreover, we achieve pressure-robustness for the mEG method by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. The theoretical results are confirmed through numerical experiments with two- and three-dimensional examples.

math.NA↗