Preconditioned Reconstructed Discontinuous Approximation For Elliptic Interface Problem on Unfitted Mesh

In this paper, we develop an efficient preconditioned unfitted finite element method for the elliptic interface problem, based on the reconstructed discontinuous approximation.

The key idea is to impose suitable constraints on the local least squares reconstruction.

These constraints ensure the stability near cut interface elements and, more importantly, establish a norm equivalence between the high-order space and the lowest-order piecewise constant space.

This result allows us to construct an optimal preconditioner directly from the lowest-order system on the same unfitted mesh for any high-order scheme.

The resulting method combines a cut discontinuous Galerkin formulation with Nitsche's penalty technique.

The approximation space achieves arbitrarily high order accuracy with only one degree of freedom per element.

We prove optimal error estimates and show that the condition number of the preconditioned system is uniformly bounded independently of the mesh size, coefficient contrast, and the location of the interface relative to the mesh.

Multigrid algorithms are further designed to efficiently approximate the inverse of the lowest-order system matrix.

Numerical experiments in two and three dimensions confirm the optimal convergence rates and demonstrate the robustness and efficiency of the proposed preconditioning method.