A scalar interface reduction for nonlinear interface problems

TL;DR

The paper introduces a scalar interface reduction method for finite element approximation of nonlinear interface problems, simplifying the nonlinear interface condition to a scalar nonlinear equation and enabling efficient computation.

Contribution

A novel scalar interface reduction technique that isolates interface nonlinearity into a single scalar variable, simplifying nonlinear interface problems.

Findings

Achieves second-order accuracy for interface quantities.

Effective computational procedure with linear solves and low-dimensional nonlinear updates.

Numerical results confirm the method's accuracy and efficiency.

Abstract

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.