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 $\kappa$. 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 $\kappa$, whereas the actual finite element solutions possess poor accuracy in a large pre-asymptotic regime which depends on $\kappa$. 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.

翻译：本文研究了Ginzburg-Landau方程（GLE）在有限元空间中的近似解性质。特别关注了网格尺寸$h$和有限元空间多项式次数$p$与Ginzburg-Landau材料参数$\kappa$的尺度耦合对误差的影响。如先前研究所示，GLE的有限元近似存在数值污染效应：在$h$与$\kappa$满足温和耦合条件时，有限元空间中的最佳逼近误差能够收敛，而实际有限元解在依赖于$\kappa$的大范围预渐近区域内精度较差。本文提出了一种新的误差分析方法，能够通过显式分辨率条件量化预渐近区域及相应的污染效应。特别地，我们证明了提高多项式次数可减轻污染效应，即在更宽松的网格尺寸条件下仍能达到最佳逼近精度。我们给出了$H^1$范数和$L^2$范数下的误差估计，并通过数值算例验证了理论结果。