Error estimates in weighted Sobolev norms for finite element immersed interface methods
Provides theoretical justification for using simpler, non-enriched immersed boundary methods while maintaining optimal global accuracy, benefiting computational scientists solving PDEs with moving interfaces.
The paper proves that for immersed interface problems solved with non-enriched methods, the suboptimal convergence is only local near the interface, and optimal global convergence can be recovered by using weighted Sobolev norms in the error measurement.
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.