arXiv subjects

Gwanghyun Jo

Publications and source records attributed to Gwanghyun Jo.

4 recordsLinked to original sources

arXiv2026

Analysis of Error Propagation in Autoencoder-Based Reduced-Order Neural Ordinary Differential Equations

Neural ODE reduced-order models often achieve comparable local prediction accuracy, yet their long-horizon extrapolation behavior can differ substantially. To analyze this discrepancy, we develop a path-integral identity that separates local discrepancy injection from amplification in the learned latent dynamics. The associated multi-step Jacobian norms quantify transport sensitivity and distinguish different propagation regimes. Experiments on the Burgers and Gray--Scott systems exhibit two distinct patterns of error evolution. In Burgers systems, prediction errors remain bounded and are primarily associated with persistent local discrepancies. In contrast, Gray--Scott systems exhibit pronounced amplification during extrapolation, where Jacobian norms serve as sensitivity diagnostics rather than direct indicators of physical prediction accuracy.

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arXiv2021

Mixed virtual volume methods for elliptic problems

We develop a class of mixed virtual volume methods for elliptic problems on polygonal/polyhedral grids. Unlike the mixed virtual element methods introduced in \cite{brezzi2014basic,da2016mixed}, our methods are reduced to symmetric, positive definite problems for the primary variable without using Lagrangian multipliers. We start from the usual way of changing the given equation into a mixed system using the Darcy's law, $\bu=-{\cal K} \nabla p$. By integrating the system of equations with some judiciously chosen test spaces on each element, we define new mixed virtual volume methods of all orders. We show that these new schemes are equivalent to the nonconforming virtual element methods for the primal variable $p$. Once the primary variable is computed solving the symmetric, positive definite system, all the degrees of freedom for the Darcy velocity are locally computed. Also, the $L^2$-projection onto the polynomial space is easy to compute. Hence our work opens an easy way to compute Darcy velocity on the polygonal/polyhedral grids. For the lowest order case, we give a formula to compute a Raviart-Thomas space like representation which satisfies the conservation law. An optimal error analysis is carried out and numerical results are presented which support the theory.

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arXiv2021

Locally conservative immersed finite element method for elliptic interface problems

In this paper, we introduce the locally conservative enriched immersed finite element method (EIFEM) to tackle the elliptic problem with interface. The immersed finite element is useful for handling interface with mesh unfit with the interface. However, all the currently available method under IFEM framework may not be designed to consider the flux conservation. We provide an efficient and effective remedy for this issue by introducing a local piecewise constant enrichment, which provides the locally conservative flux. We have also constructed and analyzed an auxiliary space preconditioner for the resulting system based on the application of algebraic multigrid method. The new observation in this work is that by imposing strong Dirichlet boundary condition for the standard IFEM part of EIFEM, we are able to remove the zero eigen-mode of the EIFEM system while still imposing the Dirichlet boundary condition weakly assigned to the piecewise constant enrichment part of EIFEM. A couple of issues relevant to the piecewise constant enrichment given for the mesh unfit to the interface has been discussed and clarified as well. Numerical tests are provided to confirm the theoretical development.

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arXiv2015

An IMPES scheme for a two-phase flow in heterogeneous porous media using a structured grid

We develop a numerical scheme for a two-phase immiscible flow in heterogeneous porous media using a structured grid finite element method, which have been successfully used for the computation of various physical applications involving elliptic equations \cite{li2003new, li2004immersed, chang2011discontinuous, chou2010optimal, kwak2010analysis}. The proposed method is based on the implicit pressure-explicit saturation procedure. To solve the pressure equation, we use an IFEM based on the Rannacher-Turek \cite{rannacher1992simple} nonconforming space, which is a modification of the work in \cite{kwak2010analysis} where `broken' $P_1$ nonconforming element of Crouzeix-Raviart \cite{crouzeix1973conforming} was developed. For the Darcy velocity, we apply the mixed finite volume method studied in \cite{chou2003mixed, kwak2010analysis} on the basis of immersed finite element method (IFEM). In this way, the Darcy velocity of the flow can be computed cheaply (locally) after we solve the pressure equation. The computed Darcy velocity is used to solve the saturation equation explicitly. Thus the whole procedure can be implemented in an efficient way using a structured grid which is independent of the underlying heterogeneous porous media. Numerical results show that our method exhibits optimal order convergence rates for the pressure and velocity variables, and suboptimal rate for saturation.

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