A Macroscopic Model for an Intermediate State Between Type-I and Type-II Superconductivity

A vectorial nonlocal and nonlinear parabolic problem on a bounded domain for an intermediate state between type-I and type-II superconductivity is proposed. The domain is for instance a multiband superconductor that combines the characteristics of both types. The nonlocal term is represented by a (space) convolution with a singular kernel arising in Eringen’s model. The nonlinearity is coming from the power law relation by Rhyner. The well-posedness of the problem is discussed under low regularity assumptions and the error estimate for a semi-implicit time-discrete scheme based on backward Euler approximation is established. In the proofs, the monotonicity methods and the Minty–Browder argument are used. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1551–1567, 2015