A scalar interface reduction for nonlinear interface problems
This work provides an efficient computational approach for solving nonlinear interface problems, which are important in applications like multiphase flow and materials science, but the method is domain-specific and incremental.
The paper introduces a scalar interface reduction method for elliptic and parabolic interface problems with discontinuous coefficients and nonlinear jump conditions, achieving second-order accuracy for interface quantities and enabling a simple computational procedure of linear solves combined with a low-dimensional nonlinear update.
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.