arXiv subjects

So-Hsiang Chou

Publications and source records attributed to So-Hsiang Chou.

11 recordsLinked to original sources

arXiv2026

Second-Order Accuracy from Large Local Errors in Interface Problems: A Discrete Green's Function Analysis

Finite difference methods for interface problems often exhibit large local truncation errors near the interface, particularly when discontinuities in coefficients or solution derivatives are present. Nevertheless, many such schemes achieve second-order accuracy, a phenomenon not fully explained by standard pointwise consistency arguments. In this work, we provide a precise explanation of this behavior through the structure of the discrete Green's function associated with the underlying conservative difference operator. An explicit representation of the Green's function reveals a two-plateau structure in its weighted increments. By expressing the numerical error in terms of this Green kernel, we show that the dominant interface truncation errors, although individually of order $O(1)$, possess a cancellation structure that reduces their effective contribution to the global error. The exact Green's-function analysis leads to a class of conservative interface flux--balance schemes for which the cancellation mechanism yields second-order accuracy in the maximum norm. The weighted harmonic discretization is included as a particular member, and its cancellation property is established directly. For Cartesian grids with a flat interface parallel to a coordinate direction, the same conservative cancellation mechanism persists along grid lines crossing the interface. Numerical experiments in one and two dimensions directly illustrate the predicted cancellation behavior and the resulting second-order accuracy. These results demonstrate that second-order accuracy in interface problems can arise from the interaction between localized truncation errors and the global structure of the discrete operator, rather than from pointwise consistency alone.

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arXiv2026

An Immersed Interface Method for Parabolic Interface Problems with Nonlinear Jump Conditions

We develop an immersed interface finite difference method for a one-dimensional nonlinear parabolic interface problem with jump condition \[ [u]_α=λu^+u^-. \] The method combines a Crank--Nicolson immersed interface discretization with an $s$-parameter reduction of the nonlinear interface condition, thereby reducing the nonlinear coupling to a scalar quadratic equation. We also discuss two different viewpoints for combining Newton iteration with immersed interface discretization, namely the IIM--Newton and Newton--IIM formulations. Numerical experiments are presented to illustrate the behavior and accuracy of the method.

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arXiv2026

Raviart--Thomas Elements with Geometric Correction for Distorted Quadrilateral Meshes

Mixed finite element methods based on Raviart--Thomas spaces are widely used for the numerical approximation of second--order elliptic problems in flux form. On quadrilateral meshes, however, the bilinear mapping from the reference element introduces a spatially varying Jacobian, which may violate the inclusion property $\mathrm{div}\,V_h \subset W_h$ for the standard Raviart--Thomas spaces. In this paper we propose a simple modification of the classical Raviart--Thomas elements on quadrilateral meshes. The modification consists of adding geometrically motivated correction terms to the local basis functions in order to compensate for the geometric distortion introduced by the bilinear mapping. The resulting spaces retain the same dimension and degrees of freedom as the classical Raviart--Thomas elements while restoring the compatibility property. We present a general framework for constructing such modified spaces and illustrate the approach by developing modified versions of the lowest order and next--to--lowest order Raviart--Thomas elements. Theoretical analysis establishes optimal approximation properties under the standard shape--regularity assumption for quadrilateral meshes. Numerical experiments on distorted meshes confirm the predicted convergence rates and show that the modified elements yield consistently improved accuracy over the classical Raviart--Thomas formulation as the geometric distortion increases.

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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.

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arXiv2026

A scalar interface reduction for nonlinear interface problems

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.

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arXiv2026

A Lifting-Based Interface Reduction Framework for Nonlinear Transmission and Eigenvalue Problems

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.

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arXiv2026

Interface Reduction for Elliptic Interface Problems with Conservative Flux Reconstruction

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.

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arXiv2023

High-Order Enriched Finite Element Methods for Elliptic Interface Problems with Discontinuous Solutions

Elliptic interface problems whose solutions are $C^0$ continuous have been well studied over the past two decades. The well-known numerical methods include the strongly stable generalized finite element method (SGFEM) and immersed FEM (IFEM). In this paper, we study numerically a larger class of elliptic interface problems where their solutions are discontinuous. A direct application of these existing methods fails immediately as the approximate solution is in a larger space that covers discontinuous functions. We propose a class of high-order enriched unfitted FEMs to solve these problems with implicit or Robin-type interface jump conditions. We design new enrichment functions that capture the imposed discontinuity of the solution while keeping the condition number from fast growth. A linear enriched method in 1D was recently developed using one enrichment function and we generalized it to an arbitrary degree using two simple discontinuous one-sided enrichment functions. The natural tensor product extension to the 2D case is demonstrated. Optimal order convergence in the $L^2$ and broken $H^1$-norms are established. We also establish superconvergence at all discretization nodes (including exact nodal values in special cases). Numerical examples are provided to confirm the theory. Finally, to prove the efficiency of the method for practical problems, the enriched linear, quadratic, and cubic elements are applied to a multi-layer wall model for drug-eluting stents in which zero-flux jump conditions and implicit concentration interface conditions are both present.

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arXiv2022

Construction of Discontinuous Enrichment Functions for Enriched FEM's for Interface Elliptic Problems in 1D

We introduce an enriched unfitted finite element method to solve 1D elliptic interface problems with discontinuous solutions, including those having implicit or Robin-type interface jump conditions. We present a novel approach to construct a one-parameter family of discontinuous enrichment functions by finding an optimal order interpolating function to the discontinuous solutions. In the literature, an enrichment function is usually given beforehand, not related to the construction step of an interpolation operator. Furthermore, we recover the well-known continuous enrichment function when the parameter is set to zero. To prove its efficiency, the enriched linear and quadratic elements are applied to a multi-layer wall model for drug-eluting stents in which zero-flux jump conditions and implicit concentration interface conditions are both present.

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arXiv2015

Flux Recovery and Superconvergence of Quadratic Immersed Interface Finite Elements

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.

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arXiv2013

Optimal Order Convergence Implies Numerical Smoothness

It is natural to expect the following loosely stated approximation principle to hold: a numerical approximation solution should be in some sense as smooth as its target exact solution in order to have optimal convergence. For piecewise polynomials, that means we have to at least maintain numerical smoothness in the interiors as well as across the interfaces of cells or elements. In this paper we give clear definitions of numerical smoothness that address the across-interface smoothness in terms of scaled jumps in derivatives [9] and the interior numerical smoothness in terms of differences in derivative values. Furthermore, we prove rigorously that the principle can be simply stated as numerical smoothness is necessary for optimal order convergence. It is valid on quasi-uniform meshes by triangles and quadrilaterals in two dimensions and by tetrahedrons and hexahedrons in three dimensions. With this validation we can justify, among other things, incorporation of this principle in creating adaptive numerical approximation for the solution of PDEs or ODEs, especially in designing proper smoothness indicators or detecting potential non-convergence and instability.

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