# Error estimates in weighted Sobolev norms for finite element immersed interface methods

## TL;DR

This paper investigates error estimates for immersed interface methods in weighted Sobolev norms, demonstrating that suboptimal convergence near interfaces is a local issue and can be mitigated with a distributionally consistent reformulation.

## Contribution

It proves that the poor convergence near interfaces is local and shows how weighted norms and reformulation improve global error estimates in immersed boundary methods.

## Key findings

- Error estimates are valid globally when using weighted norms.
- Local deterioration of convergence is confined near the interface.
- Reformulating the problem improves approximation accuracy.

## Abstract

When solving elliptic partial differential equations in a region containing immersed interfaces (possibly evolving in time), it is often desirable to approximate the problem using an independent background discretisation, not aligned with the interface itself. Optimal convergence rates are possible if the discretisation scheme is enriched by allowing the discrete solution to have jumps aligned with the surface, at the cost of a higher complexity in the implementation. A much simpler way to reformulate immersed interface problems consists in replacing the interface by a singular force field that produces the desired interface conditions, as done in immersed boundary methods. These methods are known to have inferior convergence properties, depending on the global regularity of the solution across the interface, when compared to enriched methods. In this work we prove that this detrimental effect on the convergence properties of the approximate solution is only a local phenomenon, restricted to a small neighbourhood of the interface. In particular we show that optimal approximations can be constructed in a natural and inexpensive way, simply by reformulating the problem in a distributionally consistent way, and by resorting to weighted norms when computing the global error of the approximation.