An Immersed Linear Finite Element Method with Interface Flux Capturing Recovery

A flux recovery technique is introduced for the computed solution of an immersed finite element method for one dimensional second-order elliptic problems. The recovery is by a cheap formula evaluation and is carried out over a single element at a time while ensuring the continuity of the flux across the interelement boundaries and the validity of the discrete conservation law at the element level. Optimal order rates are proved for both the primary variable and its flux. For piecewise constant coefficient problems our method can capture the flux at nodes and at the interface points exactly. Moreover, it has the property that errors in the flux are all the same at all nodes and interface points for general problems. We also show second order pressure error and first order flux error at the nodes. Numerical examples are provided to confirm the theory.