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Unconditional optimal error estimates of a modified finite element fully discrete scheme for the complex Ginzburg-Landau equation. (English) Zbl 1524.65615

Summary: In this paper, based on the 2-step backward differentiation formula (BDF2 for short) in time and the nonconforming Wilson element in space, a modified Galerkin finite element method named BDF2-MG FEM is proposed to solve the nonlinear complex Ginzburg-Landau equation (GLE for short). On one hand, a modified Ritz projection operator \(R_h\) is introduced and analyzed, which plays an important role in getting the unconditional optimal error estimates. On the other hand, a time-discrete system is constructed with the linearized BDF2 and the regularity is derived with the temporal error results. Combining these two aspects, the errors between \(R_hU^n\) and \(U_h^n\) with order \(O(h^3+h^2\triangle t)\) in \(L^2\)-norm and \(O(h^2+h^2\triangle t)\) in the modified energy norm are deduced, where \(h\) is the subdivision parameter, \(\triangle t\) is the time step, \(U^n\) and \(U_h^n\) denote the solutions of the time-discrete system and the BDF2-MG FEM respectively. Therefore the boundedness of \(\|U_h^n\|_{0,\infty}\) is proven without any restriction on the time-space grid ratio. Furthermore, by using the properties of \(R_h\), unconditional optimal error estimates of order \(O(h^3+(\triangle t)^2)\) in \(L^2\)-norm and \(O(h^2+(\triangle t)^2)\) in the modified energy norm are obtained directly. It should point out the spatial discrete errors of the BDF2-MG FEM are all one order higher than that of the BDF2 traditional Galerkin finite element method with Wilson element for the GLE. At last, a numerical experiment is presented to verify the validity of the theoretical analysis.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q56 Ginzburg-Landau equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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References:

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