Mathematical and numerical analysis of the time-dependent Ginzburg–Landau equations in nonconvex polygons based on Hodge decomposition

Abstract

Abstract. We prove well-posedness of time-dependent Ginzburg-Landau system in a nonconvex polygonal domain, and decompose the solution as a regular part plus a singular part. We see that the magnetic potential is not in H 1 in general, and the finite element method (FEM) may give incorrect solutions. To remedy this situation, we reformulate the equations into an equivalent system of elliptic and parabolic equations based on the Hodge decomposition, which avoids direct calculation of the magnetic potential. The essential unknowns of the reformulated system admit H 1 solutions and can be solved correctly by the FEMs. We then propose a decoupled and linearized FEM to solve the reformulated equations and present error estimates based on proved regularity of the solution. Numerical examples are provided to support our theoretical analysis and show the efficiency of the method.

Analysis of linearized Galerkin-mixed FEMs for the time-dependent Ginzburg-Landau equations of superconductivity

“…More important is that our analysis is given without any time-step restrictions and under more general assumptions for the regularity of the solution of the TDGL equations, which includes the problem in nonconvex polygons and certain convex polyhedrons. Usually, conventional FEMs for a scalar parabolic equation require the regularity of the solution in H 1+s with s > 0 [11,15], while for the TDGL equations in a nonconvex polygon, A ∈ H s and curl A, div A ∈ H 1+s with s < 1 in general [31]. Our numerical results show clearly that the mixed method converges to a true solution for problems in a nonconvex polygon and conventional Lagrange type FEMs do not in this case.…”

“…The above theorem shows that the convergence rate of the mixed method depends upon the order of the FEM spaces and also the regularity of the exact solution. In [31], the authors proved that on a nonconvex polygon, the TDGL equations possess regularity only with…”

“…Example 5.2 In this example, we investigate the TDGL equations (5.1)-(5.2) on an L-shape domain with a nonsmooth solution by the Galerkin-mixed method. This example was studied in [31] by a projected FEM based on Hodge decomposition. Here, we use the same exact solution as in [31], i.e., κ = 10, and the functions f , g, H e = curl A, ψ 0 and A 0 are chosen correspondingly to the following ψ and A…”

A linearized energy–conservative finite element method for the nonlinear Schrödinger equation with wave operator

“…Comparing to previous error analysis with conditional stability in [34][35][36], optimal error estimates were obtained unconditionally in [37,38]. Recently, this new technique has been used to analyze linearized FEMs for nonlinear Schrödinger type equation [24][25][26][27][28] and many other PDEs [39][40][41][42][43][44]. To our best knowledge, this new technique is mainly carried out for nonlinear parabolic type of equations and also possibly coupled with elliptic type of equations.…”

Global well-posedness of the time-dependent Ginzburg–Landau superconductivity model in curved polyhedra

“…It is noted that the standard finite element solution of the TDGL may converge to an incorrect solution, while the finite element solution of the new equivalent system converges to the true solution. Unfortunately, the new equivalent system introduced in [19,20] cannot be extended to the three-dimensional model, and the regularity of the solution in a three-dimensional curved polyhedron is even weaker than the two-dimensional solutions. In this paper, we prove existence of weak solutions for the initial-boundary value problem (1.3)-(1.6) in a general three-dimensional curved polyhedron based on the weaker embedding inequality A L 3+δ (Ω) ≤ C A Hn(curl,div) , by constructing approximating solutions which preserve the physical property 0 ≤ |ψ| ≤ 1.…”