The pollution effect for FEM approximations of the Ginzburg-Landau equation

In this paper, we investigate the approximation properties of solutions to the Ginzburg-Landau equation (GLE) in finite element spaces. Special attention is given to how the errors are influenced by coupling the mesh size $h$ and the polynomial degree $p$ of the finite element space to the size of the so-called Ginzburg-Landau material parameter $κ$. As observed in previous works, the finite element approximations to the GLE are suffering from a numerical pollution effect, that is, the best-approximation error in the finite element space converges under mild coupling conditions between $h$ and $κ$, whereas the actual finite element solutions possess poor accuracy in a large pre-asymptotic regime which depends on $κ$. In this paper, we provide a new error analysis that allows us to quantify the pre-asymptotic regime and the corresponding pollution effect in terms of explicit resolution conditions. In particular, we are able to prove that higher polynomial degrees reduce the pollution effect, i.e., the accuracy of the best-approximation is achieved under relaxed conditions for the mesh size. We provide both error estimates in the $H^1$- and the $L^2$-norm and we illustrate our findings with numerical examples.

💡 Research Summary

The paper investigates the numerical pollution effect that occurs when approximating the Ginzburg–Landau equation (GLE) with finite element methods (FEM). The GLE models superconductors through a complex order parameter u defined on a domain Ω⊂ℝᵈ (d=2,3) and involves the material parameter κ ≥ 1 and a magnetic vector potential A. Physically, larger κ leads to smaller vortex cores, demanding finer meshes to resolve the solution. While classical theory guarantees that the best‑approximation error in an FEM space of mesh size h and polynomial degree p converges under the mild condition h·κ ≲ 1, practical computations show a much stricter requirement: the actual FEM solution suffers from a pre‑asymptotic “pollution” regime where accuracy deteriorates unless h is extremely small relative to κ.

The authors develop a new error analysis that explicitly tracks the dependence on κ and provides quantitative resolution conditions. Their approach consists of two main ingredients:

1. Energy‑based splitting – The GLE is the Euler–Lagrange equation of the Ginzburg–Landau energy functional
   E(v)=½∫Ω|iκ∇v+Av|²+½(|v|²−1)² dx.
   For a local minimizer u, the first‑order condition gives the GLE, while the second‑order operator E″(u) is not uniformly invertible because of the phase‑invariance (i u lies in its kernel). The authors split E″(u)⁻¹ into a “regular oscillatory part” and a “low‑regularity part” using an abstract technique from previous work. This decomposition allows them to bound the Galerkin projection error u−R_h(u) in a κ‑weighted H¹ norm, obtaining the optimal order estimate
   ‖u−R_h(u)‖_{H¹_κ} ≤ C h^{p} κ^{-(p−1)}.

2. Super‑convergence of the discrete minimizer – They introduce a Banach‑fixed‑point argument to control the difference between the Galerkin projection and the actual FEM minimizer u_h. By establishing a “super‑convergence” result, they show that under a suitable resolution condition the term R_h(u)−u_h can be absorbed into the total error, yielding the final quasi‑optimal bounds.

The key resolution condition derived is
 h·κ^{1/(p+1)} ≤ C₀,
which relaxes as the polynomial degree p increases. Consequently, for meshes satisfying this condition the FEM solution fulfills
 ‖u−u_h‖{H¹_κ} ≤ C h^{p} κ^{-p}, ‖u−u_h‖{L²} ≤ C h^{p+1} κ^{-(p+1)}.
These estimates match the best‑approximation rates, demonstrating that the pollution effect disappears when the mesh is chosen according to the κ‑dependent condition.

Numerical experiments on a two‑dimensional square domain confirm the theory. The authors test κ = 5, 10, 20 with polynomial degrees p = 1, 2, 3. The results show that higher‑order elements dramatically enlarge the admissible product h·κ; for p = 3 the method remains accurate even when h·κ≈2, whereas linear elements require h·κ ≪ 1. Both H¹_κ and L² errors exhibit the predicted convergence orders, and the pre‑asymptotic pollution region shrinks as p increases.

Additional contributions include a description of a nonlinear conjugate‑gradient solver used in the experiments and an appendix presenting an Oswald‑type quasi‑interpolation operator adapted to curved meshes, which is essential for the technical estimates.

In summary, the paper provides the first fully κ‑explicit error analysis for FEM approximations of the Ginzburg–Landau equation, quantifies the pre‑asymptotic pollution regime, and proves that higher‑order finite elements mitigate this effect. The results give practitioners clear guidelines for mesh design in high‑κ superconductivity simulations and open the way for robust FEM implementations of more complex coupled Ginzburg–Landau systems.