Nonconforming $hp$-FE/BE coupling on unstructured meshes based on Nitsche's method

This paper presents and analyzes a stable $hp$-finite element/boundary element coupling method on non-matching unstructured meshes using Nitsche's method, which avoids the Babuška-Brezzi condition by providing explicit stabilization thresholds and a priori error estimates for both bounded and unbounded domain configurations.

The Gist

Imagine you are trying to solve a giant, complex puzzle, but you have two different teams working on different parts of it.

• Team A (The Finite Elements) works on the inside of the puzzle. They use a grid of tiny, flexible tiles to map out the details. They can make the tiles smaller or larger depending on how much detail is needed.
• Team B (The Boundary Elements) works on the outside edge of the puzzle. Instead of filling the whole space, they just trace the perimeter, like drawing a line around the shape.

The problem arises when these two teams try to meet in the middle. In the past, they had to agree on a very strict rule: their grid lines had to match perfectly where they touched. If Team A had small tiles and Team B had long lines, they couldn't talk to each other easily. This made the puzzle very hard to solve, especially if the shape was weird or had sharp corners.

The New Solution: "Nitsche's Method"

This paper introduces a clever new way for these two teams to collaborate, even if their grids don't match at all.

Think of it like two neighbors building a fence between their yards.

• The Old Way (Mortar Method): They had to hire a strict inspector (a "Lagrange Multiplier") to stand between them and force their fence posts to line up exactly. This was expensive, complicated, and if the inspector made a mistake, the whole fence could collapse.
• The New Way (Nitsche's Method): Instead of a strict inspector, they use a smart, flexible handshake. They agree on a set of rules that say, "Hey, your side of the fence is here, and my side is there. As long as we are close enough and we both pay a little 'penalty' if we drift apart too much, we are good."

This "handshake" is mathematically called Nitsche's method. It makes the system stable and positive (meaning it always finds a solution) without needing that complicated inspector.

The "hp" Superpower

The paper also talks about something called hp-FE/BE. Let's break that down with a metaphor:

1. The 'h' (Mesh Size): This is about zooming in. If you have a blurry photo, you can make it clearer by taking more, smaller pixels. In math, this means using smaller tiles near the tricky parts of the puzzle.
2. The 'p' (Polynomial Degree): This is about smarter tiles. Instead of just using flat, square tiles, you can use tiles that are curved or wavy to fit the shape better. A higher "p" means the tiles are more intelligent and can bend to fit the curve perfectly.

The Breakthrough:
Usually, if you have a puzzle with a sharp, jagged corner (a "singularity"), just making the tiles smaller (h) isn't enough. You need to make the tiles smarter (p) right near that corner.

This paper proves that you can mix and match these two strategies freely. You can have tiny, smart tiles in one area and big, simple ones in another, even if the two teams (FE and BE) are using completely different grid sizes.

Why Does This Matter?

1. No More "Perfect Alignment": You don't need to waste time trying to make the two grids match perfectly. You can just throw them together, and the math handles the rest.
2. Handling the "Sharp Corners": Many real-world problems (like stress in a bridge or heat in an engine) have sharp corners where things get crazy. This method allows the computer to focus its power exactly where it's needed (using geometric refinement) without getting confused.
3. Speed and Accuracy: The authors show that this method doesn't just work; it works fast. It can find the answer with incredible precision, much faster than older methods, especially when dealing with difficult shapes.

The "Stabilization" Secret Sauce

There is one catch: The "handshake" needs a little bit of glue to hold it together. The paper calculates exactly how much glue (called the stabilization parameter) is needed.

• Too little glue: The fence falls over (the math becomes unstable).
• Too much glue: The fence becomes rigid and hard to move (the math becomes hard to solve).
• Just right: The paper gives a precise formula for the "just right" amount, ensuring the system is stable and accurate.

In a Nutshell

This paper is like a new instruction manual for two different construction crews working on the same building. It tells them: "You don't need to match your brick sizes. You don't need a strict foreman. Just use this specific handshake rule, and you can build a perfect, stable structure together, even if the building has weird, sharp corners."

It's a major step forward in making computer simulations of the real world faster, more flexible, and more accurate.

Technical Summary

1. Problem Statement

The paper addresses the numerical solution of interface problems and problems on unbounded domains by coupling the Finite Element Method (FEM) and the Boundary Element Method (BEM).

• Domain Decomposition: The computational domain $\Omega$ is split into a bounded subdomain $\Omega_1$ (treated with FEM) and an unbounded or exterior subdomain $\Omega_2$ (treated with BEM).
• The Challenge: The coupling interface $\Gamma_I = \partial\Omega_1 \cap \partial\Omega_2$ often involves non-matching meshes (the discretizations on either side of the interface do not align) and variable polynomial degrees (the $hp$-setting).
• Goal: To develop a stable and convergent coupling scheme that handles non-matching meshes and allows for local mesh refinement ($h$) and variable polynomial orders ($p$), particularly for problems with singular solutions (e.g., re-entrant corners).

2. Methodology

The authors propose a Nitsche-type coupling method adapted for the $hp$-framework.

A. Variational Formulation

Instead of using Lagrange multipliers (which leads to saddle-point problems requiring the inf-sup/Babuška-Brezzi condition), the method uses Nitsche's method to weakly enforce continuity across the interface.

• Weak Continuity: The continuity of the solution $u$ and the flux is enforced via penalty terms added to the bilinear form.
• Symmetric Formulation: The method employs a symmetric boundary element formulation using the Steklov-Poincaré operator ($S$) and the Newton potential ($N$).
• Discrete Operators: Since $S$ and $N$ involve non-local integrals, they are approximated using discrete operators ($\hat{S}$ and $\hat{N}$) based on $L^2$-projections of the boundary integral operators. This introduces a consistency error which is rigorously bounded.

B. The Stabilization Parameter

A critical component of Nitsche's method is the stabilization function $\eta$.

• Local Dependence: Unlike traditional methods that assume quasi-uniform meshes, this paper derives a stabilization parameter $\eta$ that depends locally on the mesh size $h_K$ and the polynomial degree $p_K$.
• Inverse Inequalities: The choice of $\eta$ is derived from local inverse inequalities. Specifically, $\eta = \eta_0 \kappa G$, where $G$ is the constant from the polynomial trace inequality (scaling as $h^{-1}p^2$) and $\eta_0$ is a sufficiently large constant.
• Positive Definiteness: This choice ensures the resulting algebraic system is positive definite, avoiding the need for mixed formulations.

C. Discretization Strategies

The method supports two types of discretizations:

1. Quasi-uniform meshes: Standard $h$ and $p$ refinements.
2. Geometrically refined meshes: Specifically designed for corner singularities. The mesh is refined geometrically towards singular points (with a grading factor $\sigma$), and the polynomial degree $p$ increases linearly with the layer number ($p \sim \mu \cdot \text{layer}$).

3. Key Contributions

A. Stability Analysis without Inf-Sup Conditions

The paper proves that the Nitsche formulation is stable and positive definite provided the stabilization parameter exceeds a specific threshold derived from local inverse inequalities. This eliminates the need for the Babuška-Brezzi condition required by mortar methods.

B. Explicit A Priori Error Estimates

The authors derive a priori error estimates that are explicit in both the local mesh size ($h$) and the polynomial degree ($p$).

• Quasi-uniform case: The error bound is optimal in $h$ and suboptimal in $p$ by a factor of $p^{1/2}$ (a standard result in Discontinuous Galerkin methods).
• Singular solutions: The analysis extends to geometrically graded meshes, proving that the method achieves exponential convergence for solutions with singularities, provided the mesh grading and polynomial growth rates are chosen correctly.

C. Handling Non-Matching Meshes

The method is robust for non-matching meshes on the interface. The analysis does not require the trace meshes of the FEM and BEM sides to be identical, only that they cover the interface.

D. Algebraic Structure

The paper details the algebraic structure of the coupled system, showing how the stiffness matrices for FEM and BEM are combined with the coupling terms (consistency and stabilization) to form a sparse (FEM part) and dense (BEM part) global system that remains positive definite.

4. Results

Theoretical Results

• Stability: Theorem 4 establishes quasi-optimal convergence in a mesh-dependent energy norm.
• Convergence Rates:
  • For smooth solutions, the method achieves optimal algebraic convergence rates in $h$ and exponential convergence in $p$.
  • For singular solutions (e.g., $r^{2/3}$ in L-shaped domains), the $hp$-method with geometric refinement achieves exponential convergence with respect to the square root of the degrees of freedom ($N^{1/2}$ or $N^{1/3}$ depending on the domain split).

Numerical Experiments

The authors validate the theory with numerical examples:

1. Smooth Solution (Square Domain):
   • Demonstrated exponential convergence for the $p$-version.
   • Confirmed optimal convergence rates for the $h$-version with varying $p$.
2. Singular Solution (L-shaped Domain):
   • Configuration 1: The singularity is entirely within the BEM domain. The method showed exponential convergence.
   • Configuration 2: The singularity is split between FEM and BEM domains (at the interface). The method still achieved exponential convergence, though the scaling with degrees of freedom was slightly different due to the dominance of FEM degrees of freedom in the refined region.
   • The numerical results confirmed the theoretical prediction that the method is robust even when the singularity lies on the non-matching interface.

5. Significance

• Robustness for Complex Geometries: The method provides a reliable way to couple FEM and BEM on unstructured, non-matching meshes, which is crucial for complex engineering problems where different parts of a domain require different discretization strategies.
• Efficiency for Singularities: By enabling $hp$-refinement with geometric grading, the method offers a highly efficient approach for problems with singularities (common in fracture mechanics, electromagnetics, and fluid dynamics), achieving exponential convergence where standard $h$-FEM would struggle.
• Simplicity of Implementation: Unlike mortar methods, this approach does not require the construction of Lagrange multiplier spaces or the satisfaction of complex inf-sup conditions, making it easier to implement in existing FEM/BEM codes while maintaining stability.
• Theoretical Rigor: The paper fills a gap in the literature by providing the first sharp, local stability estimates and error bounds for Nitsche-based FE/BE coupling that are explicit in both $h$ and $p$, specifically addressing the challenges of non-uniform meshes.

In summary, this work establishes a mathematically rigorous and computationally efficient framework for solving interface problems using Nitsche's method, successfully extending the technique to the high-order ($hp$) regime on non-matching meshes.