A Hodge Decomposition Method for Dynamic Ginzburg–Landau Equations in Nonsmooth Domains — A Second Approach (Communications in Computational Physics, 2020)
In a general polygonal domain, possibly nonconvex and multi-connected(with holes), the time-dependent Ginzburg–Landau equation is reformulated into anew system of equations. The magnetic field $B$:=∇×A is introduced as an unknownsolution in the new system, while the magnetic potential A is solved implicitly throughits Hodge decomposition into divergence-free part, curl-free and harmonic parts, separately. Global well-posedness of the new system and its equivalence to the original problem are proved. A linearized and decoupled Galerkin finite element methodis proposed for solving the new system. The convergence of numerical solutions isproved based on a compactness argument by utilizing the maximal $L^p$-regularity ofthe discretized equations. Compared with the Hodge decomposition method proposed in [27], the new method has the advantage of approximating the magnetic fieldB directly and converging for initial conditions that are incompatible with the externalmagnetic field. Several numerical examples are provided to illustrate the efficiency ofthe proposed numerical method in both simply connected and multi-connected nonsmooth domains. We observe that even in simply connected domains, the new methodis superior to the method in [27] for approximating the magnetic field.
Keywords: Superconductivity; reentrant corner; singularity; multi-connected domain; finite element method; convergence; Hodge decomposition.