Abstract

The real Ginzburg–Landau equation is studied in this article. We proceed and use the Galerkin method in combination with the compactness theorem to show that the solution of this problem exists and is unique in appropriate Sobolev spaces. This is proceeded by the designing of a reliable nonlinear numerical scheme from the afore mentioned problem and further show that this scheme is stable. Furthermore, the optimal rate of convergence of the scheme is determined in some appropriate spaces with emphasizes on the fact that the numerical solution from this scheme preserves all the qualitative properties of the exact solution and the numerical experiments are conducted with the help of an example to justify the theory.

摘要

本文研究了真正的 Ginzburg-Landau 方程。我们继续并使用Galerkin方法结合紧性定理来证明该问题的解存在并且在适当的Sobolev空间中是唯一的。这是通过从上述问题设计可靠的非线性数值方案进行的，并进一步表明该方案是稳定的。此外，该方案的最佳收敛速度是在一些适当的空间确定的，并强调该方案的数值解保留了精确解的所有定性性质，并借助一个例子进行了数值实验证明理论。