So-Hsiang Chou

So-Hsiang Chou

We present a lifting-based interface reduction framework for nonlinear transmission and eigenvalue problems. The method represents the solution as a sum of a bulk component and a lifting component that carries the interface jump, thereby reducing the original problem to a nonlinear system posed on the interface. A low-dimensional approximation is obtained by restricting the interface unknown to a finite-dimensional subspace. The corresponding lifting modes are precomputed and reused, leading to a formulation in which the bulk operator remains fixed and the essential behavior is governed by a small number of interface degrees of freedom. For eigenvalue problems, the same framework yields a reduced system in which eigenvalues are identified through the near-singularity of a parameter-dependent interface matrix. The associated eigenvectors reveal dominant interface modes, providing a direct interpretation of the spectral structure. Numerical experiments show that both approximation accuracy and spectral behavior are determined primarily by the interface representation rather than by the bulk discretization. In particular, enriching the interface space rapidly improves accuracy and reveals additional eigenmodes, while mesh refinement alone has limited effect. These results indicate that transmission and eigenvalue problems are effectively governed by a small number of interface modes, offering a simple and computationally efficient perspective on model reduction.

So-Hsiang Chou, C. Attanayake

We introduce a flux recovery scheme for the computed solution of a quadratic immersed finite element method. The recovery is done at nodes and interface point first and by interpolation at the remaining points. We show that the end nodes are superconvergence points for both the primary variable $p$ and its flux $u$. Furthermore, in the case of piecewise constant diffusion coefficient without the absorption term the errors at end nodes and interface point in the approximation of $u$ and $p$ are zero. In the general case, flux error at end nodes and interface point is third order. Numerical results are provided to confirm the theory.

C. Attanayake, So-Hsiang Chou

We propose a low-dimensional interface reduction method for elliptic interface problems based on conservative flux reconstruction. The approach combines a fitted $P_1$ finite element discretization with a flux recovery procedure following \cite{ChouTang2000}, yielding locally conservative fluxes that satisfy interface conditions to machine precision. A central result shows that the error of the reduced solution is controlled entirely by the approximation error of the interface data. Numerical experiments for both continuous and discontinuous interface conditions confirm that once the interface data is accurately represented, the full solution is recovered to roundoff accuracy. These results indicate that the essential complexity of elliptic interface problems is concentrated on the interface.

So-Hsiang Chou

We study finite element approximations of elliptic and parabolic interface problems with discontinuous coefficients and nonlinear jump conditions. We introduce a scalar interface reduction in which the solution is decomposed into a continuous component and a unit-jump response mode. This representation isolates the interface nonlinearity into a single scalar variable while the bulk problem remains linear. From this perspective, the nonlinear interface condition is reduced to a scalar nonlinear equation, which may be interpreted as a nonlinear Schur complement associated with the interface degree of freedom. The resulting formulation leads to a simple computational procedure consisting of linear solves combined with a low-dimensional nonlinear update. Numerical results for representative elliptic and parabolic problems confirm second-order accuracy for interface quantities and demonstrate the effectiveness of the proposed approach.