Superconvergence of Immersed Finite Element Methods for Interface Problems
This provides a theoretical foundation for superconvergence in immersed finite element methods, benefiting researchers working on numerical methods for interface problems.
The paper proves that immersed finite element methods for 1D elliptic interface problems achieve superconvergence at roots of generalized orthogonal polynomials, without requiring mesh alignment with the interface, overcoming the loss of superconvergence due to low global regularity.
In this article, we study superconvergence properties of immersed finite element methods for the one dimensional elliptic interface problem. Due to low global regularity of the solution, classical superconvergence phenomenon for finite element methods disappears unless the discontinuity of the coefficient is resolved by partition. We show that immersed finite element solutions inherit all desired superconvergence properties from standard finite element methods without requiring the mesh to be aligned with the interface. In particular, on interface elements, superconvergence occurs at roots of generalized orthogonal polynomials that satisfy both orthogonality and interface jump conditions.