The New Adaptive Finite Element Method For The Time-Dependent Ginzburg-Landau Equations

In this paper,we mainly focus on the theoretical analysis of a new residual-based type a posteriori error estimation for a new finite element method about the timedependent Ginzburg-Landau(TDGL)equation arising from the superconductivity and present the numerical results of the adaptive algorithm,and then we study a new stabilized mixed finite element method for solving the Maxwell eigenvalue problem.1)For two-dimensional and three-dimensional TDGL superconducting equations,we propose a new finite element method.For magnetic potential variable A,we choose H(▽×;Ω)-conforming element such that when A has Hr(0<r<1)low regularity H(▽×;Ω)element can also approximate the numerical solution well;For the complex-valued order parameter ψ,considering that ψ also has low regularity,such as H1+s(0<s<1),we select H1(Ω)conforming element to obtain the correct numerical solution.The low regularity of variables A and ψ is usually generated by re-entrant corners and edges in a non-smooth domain.For solutions with low regularity,the optimal error estimate of high-order finite element space cannot be obtained even if we choose high-order finite element space in the procedure of finite element discretization.Therefore,in order to deal with solutions with low regularity and non-smooth initial value conditions of L2 only,we propose a new posterior error estimator based on residual errors.Since the TDGL equation is nonlinear,we first linearize it,and analyze the dual problem of the linearized equation.Then we prove the stability of the dual linear problem rigorously.The a posterior error estimator of the residual type is derived and the reliability analysis of the posterior error estimator is given.According to the constructed posterior error estimator,an adaptive algorithm is designed to refine and coarsen in time and space.Numerical examples show that the residual-type error estimator constructed can accurately indicate the subregions to be processed in time and space,and the adaptive finite element method is also very effective for singular solutions and non-convex domains.When the regularity of the solution is low,the optimal error order of discrete finite element space can also be obtained by adaptive algorithm.2)For the Maxwell eigenvalue problem,a Bochev-Dohrmann-Gunzburger stabilization mixed finite element method based on piecewise constant projection is presented,and some ad hoc stabilizing parameters are added to the stabilization method.According to the general theory of the mixed method corresponding to the saddle-point problem related to the kernel-coercivity and inf-sup condition,for some edge elements that do not satisfy the discrete exact sequence of de Rham complex,that is,these elements lose stability and convergence during numerical calculation,and produce incorrect discrete eigenvalues,by stabilizing the kernelcoercivity and inf-sup conditions with the stabilization term,the spectral-correct and spurious-free of the Maxwell discrete eigenvalues are obtained.The optimal error result and relationship between the stabilization parameters is obtained by Babu?ka-Osborn spectrum theory.We prove the optimal convergence,up to an arbitrarily small constant for the discrete eigenmodes.The effectiveness of the new stabilized mixed finite element method and the theoretical results are also verified by numerical experiments.