The Tempered Finite Element Method

Technical Summary: The Tempered Finite Element Method (TFEM)

Problem Statement
The Finite Element Method (FEM) is the dominant numerical technique in computational physics and engineering, renowned for its ability to handle complex geometries via unstructured meshes. However, standard FEM formulations exhibit brittleness when encountering specific mesh configurations, particularly those containing "caps" (triangles with an angle approaching 180°) arranged in bands. While isolated degenerate elements (needles or isolated caps) do not prevent convergence, bands of caps induce a "locking" phenomenon. In such configurations, the numerical solution is forced to be linear along the band regardless of the true solution, causing the effective mesh size to be determined by the band's length rather than its thickness. This results in a loss of convergence for standard FEM. Historically, the "maximum angle condition" (requiring all angles to be bounded away from 180°) was considered essential for convergence, though recent work has shown it is not strictly necessary for isolated elements. The challenge remains to handle meshes with zero-measure or nearly degenerate elements, which are relevant for interface tracking methods (e.g., X-Mesh) where interfaces are tracked by mesh relaying, potentially creating bands of caps.

Methodology
The authors propose the Tempered Finite Element Method (TFEM), a modification of the classical FEM that allows convergence on meshes containing bands of caps. The core idea is inspired by metallurgical tempering: just as tempering relieves internal stresses in hardened steel to increase flexibility, TFEM locally "softens" the material properties in the degenerate band to allow the numerical solution to conform to the continuous problem.

Mathematically, for elliptic problems like the Poisson equation, the standard stiffness matrix $K_{ij}$ involves the inverse of the Jacobian determinant $J$ of the element mapping. In degenerate elements, $J \to 0$, leading to division by zero or infinite eigenvalues. TFEM modifies the local stiffness matrix by bounding the Jacobian away from zero using a parameter $J_{min}$:
$$\tilde{K}_{ij} = \int_{\hat{E}} \frac{1}{\max(J, J_{min})} [\dots] d\xi$$
This is equivalent to introducing a variable diffusion coefficient $d(x)$ in the weak formulation:
$$d(x) = \begin{cases} 1 & x \in \Omega \setminus B \\ D = \frac{J}{J_{min}} & x \in B \end{cases}$$
where $B$ is the band of degenerate elements. The parameter $J_{min}$ is chosen to be dependent on the mesh size $h$ (specifically $J_{min} \sim h^{d+1}$ where $d$ is the spatial dimension) to balance the reduction of the diffusion coefficient (to avoid locking) with the preservation of approximation properties.

For the case of exactly zero-measure elements ($J=0$), the authors demonstrate that the limit of the TFEM scheme corresponds to a mortaring method with penalization of the solution jump across the interface.

Key Contributions

1. Theoretical Proof of Convergence: The paper provides a rigorous error analysis for TFEM on bands of caps. It proves that with an optimal choice of the tempering parameter $D \sim \bar{h}/h^2$ (where $\bar{h}$ is the band width), the method achieves optimal $O(h)$ convergence in the $H^1$ semi-norm, matching the rate of standard FEM on regular meshes.
2. Equivalence to Mortaring: For exactly zero-width bands, the authors prove that TFEM is equivalent to a non-conforming mortar method with jump penalization, providing a theoretical bridge between degenerate mesh handling and interface coupling techniques.
3. Extension to Complex Physics: The method is successfully extended beyond the scalar Laplacian to:
   • Linear Elasticity: Demonstrating comparable accuracy to standard FEM for stress analysis across interfaces.
   • Non-conforming Mesh Mortaring: Showing that TFEM can seamlessly connect meshes of different resolutions without transitional elements.
   • High-Order Elements: Validating that the method preserves optimal convergence rates for polynomial orders up to 4, provided $J_{min}$ is scaled appropriately with the element order.
   • Advection: Applying TFEM to advection-diffusion problems using Streamline Upwind/Petrov-Galerkin (SUPG) stabilization, noting that the tempering is only required for the stabilization term.
4. Robustness Analysis: Numerical experiments identify that the method is robust with respect to the constant $C$ in the relation $J_{min} = C h^{d+1}$. The error remains close to optimal over a wide range of $C$ values (spanning nearly an order of magnitude), and the conditioning of the linear system remains manageable, especially with standard preconditioners.

Results

• Numerical Experiments: In 2D and 3D Poisson problems with manufactured solutions, TFEM on meshes with bands of caps recovers the convergence rates of standard FEM on regular meshes. Specifically, choosing $J_{min} \sim h^3$ in 2D and $J_{min} \sim h^4$ in 3D yields optimal $L^2$ and $H^1$ error convergence.
• Error Sensitivity: The $H^1$ error is less sensitive to the choice of $J_{min}$ than the $L^2$ error, exhibiting a plateau over a wide range of parameters.
• Conditioning: While the condition number of the stiffness matrix increases as the band width decreases, the use of incomplete Cholesky preconditioning ensures that iterative solver convergence remains independent of the band thickness.
• Randomized Testing: A suite of 100 random tests in 2D confirmed that even with a simple choice ($C=1$), TFEM errors are nearly identical to standard FEM errors on regular meshes.

Significance and Claims
The paper claims that TFEM significantly increases the flexibility of finite element analysis by relaxing the strict mesh quality requirements associated with degenerate elements. It allows for the use of meshes with zero-measure or nearly degenerate elements (such as those generated by dynamic interface tracking) without suffering from the locking phenomena that render standard FEM inoperative.

The authors emphasize that the method is trivial to implement in existing FEM codes, requiring only a modification to the Jacobian evaluation to prevent division by zero. They position TFEM not as a replacement for standard FEM on high-quality meshes, but as a robust extension that enables the use of specific mesh topologies (bands of caps) that were previously considered inoperative. The method is particularly valuable for applications like the X-Mesh interface tracking method, where maintaining a conforming mesh is difficult or computationally expensive. The paper concludes that while the theoretical optimal choice of the parameter $C$ is derived, the method's practical robustness allows for a wide range of acceptable values, making it immediately applicable.