Robust high-order unfitted finite elements by interpolation-based discrete extension

TL;DR

This paper introduces a robust high-order unfitted finite element method using interpolation-based discrete extension, improving stability and convergence for PDEs on complex geometries.

Contribution

It develops a new variant of aggregated finite elements with polynomial interpolation extension, reducing stability constant growth from exponential to polynomial rate.

Findings

Enhanced stability on the entire active mesh.

Optimal convergence rates achieved.

Condition number scales optimally with polynomial order.

Abstract

In this work, we propose a novel formulation for the solution of partial differential equations using finite element methods on unfitted meshes. The proposed formulation relies on the discrete extension operator proposed in the aggregated finite element method. This formulation is robust with respect to the location of the boundary/interface within the cell. One can prove enhanced stability results, not only on the physical domain, but on the whole active mesh. However, the stability constants grow exponentially with the polynomial order being used, since the underlying extension operators are defined via extrapolation. To address this issue, we introduce a new variant of aggregated finite elements, in which the extension in the physical domain is an interpolation for polynomials of order higher than two. As a result, the stability constants only grow at a polynomial rate with the order of approximation. We demonstrate that this approach enables robust high-order approximations with the aggregated finite element method. The proposed method is consistent, optimally convergent, and with a condition number that scales optimally for high order approximation.