Interface Reduction for Elliptic Interface Problems with Conservative Flux Reconstruction
For researchers solving elliptic interface problems, this method reduces computational complexity by focusing on the interface, but it is an incremental improvement over existing flux reconstruction techniques.
The paper introduces a low-dimensional interface reduction method for elliptic interface problems that achieves roundoff accuracy once interface data is accurately represented, demonstrating that the problem's complexity is concentrated on the interface.
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.