arXiv subjects

Do Young Kwak

Publications and source records attributed to Do Young Kwak.

3 recordsLinked to original sources

arXiv2026

A Trace-Based Interface Reduction Method for Highly Conducting Interfaces

We develop a reduced interface formulation for elliptic interface problems with highly conducting interfaces. The interface condition consists of continuity of the primal variable together with a jump in the normal flux proportional to the surface Laplacian of the interface trace. Instead of using the solution jump as the interface unknown, we employ the common interface trace and derive a trace-based Schur complement formulation. For prescribed interface trace data, independent extension problems are solved in the two subdomains, leading to a reduced interface equation involving the Dirichlet-to-Neumann jump operator and a surface stiffness operator. Finite-dimensional trace approximations produce compact reduced systems posed only on the interface. Numerical experiments for circular, smooth noncircular, and heart-shaped interfaces illustrate the effectiveness of the method and the role of interface-mode enrichment.

math.NA↗

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.

math.NA↗

arXiv2015

A generalization of the divide and conquer algorithm for the symmetric tridiagonal eigenproblem

In this paper, we present a generalized Cuppen's divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. We extend the Cuppen's work to the rank two modifications of the form $A =T +β_1\bw_1\bw_1^T + β_2\bw_2\bw_2^T$, where $T$ is a block tridiagonal matrix having three blocks. We introduce a new deflation technique and obtain a secular equation, for which the distribution of eigenvalues is nontrivial. We present a way to count the number of eigenvalues in each subinterval. It turns out that each subinterval contains either none, one or two eigenvalues. Furthermore, computing eigenvectors preserving the orthogonality are also suggested. Some numerical results, showing our algorithm can calculate the eigenvalue twice as fast as the Cuppen's divide-and-conquer algorithm, are included.

math.NA↗