C. Attanayake

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.