[Paper Review] An unfitted $hp$-interface penalty finite element method for elliptic interface problems

Haijun Wu, Yuanming Xiao|arXiv (Cornell University)|Jul 17, 2010

Advanced Numerical Methods in Computational MathematicsEngineering29 references17 citations

TL;DR

This paper proposes an $hp$-version unfitted interface penalty finite element method (hp-IPFEM) for elliptic interface problems on unfitted meshes in 2D and 3D. It employs a symmetric and non-symmetric variational formulation with interface flux stabilization and penalty terms, achieving optimal $h$-convergence and suboptimal $p$-convergence (by half an order) in the broken $H^1$ norm, with $L^2$ error estimates derived via duality argument.

ABSTRACT

An $hp$ version of interface penalty finite element method ($hp$-IPFEM) is proposed for elliptic interface problems in two and three dimensions on unfitted meshes. Error estimates in broken $H^1$ norm, which are optimal with respect to $h$ and suboptimal with respect to $p$ by half an order of $p$, are derived. Both symmetric and non-symmetric IPFEM are considered. Error estimates in $L^2$ norm are proved by the duality argument.

Motivation & Objective

• To develop a high-order, stable finite element method for elliptic interface problems on unfitted Cartesian meshes where the interface cuts through elements without requiring body-fitted triangulation.
• To achieve optimal convergence rates in the mesh size $h$ and suboptimal but sharp convergence in the polynomial degree $p$ for the broken $H^1$ norm.
• To extend the penalty finite element method to the $hp$-version framework, ensuring stability and convergence for both symmetric and non-symmetric formulations.
• To derive rigorous error estimates in both $H^1$ and $L^2$ norms using duality arguments, ensuring robustness with respect to interface location.

Proposed method

• Formulates a symmetric and non-symmetric $hp$-interface penalty finite element method (hp-IPFEM) using a variational formulation that penalizes jumps in solution and flux across the interface.
• Applies a weighted average flux technique in the bilinear form to stabilize the method, improving accuracy and conditioning on unfitted meshes.
• Introduces penalty terms for both solution and flux jumps to enforce interface conditions weakly, ensuring stability and convergence.
• Employs a discontinuous Galerkin-like approach with penalty stabilization, allowing high-order polynomial approximations ($p$-refinement) on elements cut by the interface.
• Uses a trace inequality on curved interface patches to control boundary terms, derived via cone-type geometry and parameterization of the interface.
• Applies duality arguments to derive $L^2$ error estimates, extending the convergence analysis beyond the $H^1$-norm.

Experimental results

Research questions

• RQ1 Can an $hp$-version finite element method be constructed that maintains optimal $h$-convergence and near-optimal $p$-convergence on unfitted meshes for elliptic interface problems?
• RQ2 How does the choice of symmetric vs. non-symmetric formulation affect the stability and convergence of the $hp$-IPFEM?
• RQ3 What is the optimal penalty parameter scaling to ensure stability and optimal convergence in the broken $H^1$ norm for unfitted $hp$-FEM?
• RQ4 Can the method achieve $L^2$ convergence rates comparable to standard FEMs despite the unfitted mesh and interface cuts?
• RQ5 How do geometric assumptions on the interface (e.g., $C^2$-smoothness) and mesh shape regularity affect the error estimates?

Key findings

• The method achieves optimal $h$-convergence rate in the broken $H^1$ norm, with error estimates independent of the interface location relative to the mesh.
• The $p$-convergence rate is suboptimal by half an order of $p$, i.e., $O(h^p + h^{p+1/2})$, which is the best known for unfitted $hp$-methods in this context.
• Error estimates in the $L^2$ norm are derived via duality argument, confirming convergence under standard regularity assumptions.
• The symmetric and non-symmetric formulations both yield stable and convergent schemes, with the symmetric variant offering better conditioning.
• The analysis relies on a novel trace inequality on curved interface patches, proven via cone-based geometry and parameterization, ensuring robustness for small elements.
• The method maintains uniform stability and convergence even when the interface cuts elements in arbitrary positions, avoiding the need for remeshing or complex basis function modifications.