Nonlinear diffusion in type-II superconductors

• Published in: Journal of computational and applied mathematics Vol. 215; no. 2; pp. 568 - 576
• Main Author: SLODICKA, Marian
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Amsterdam Elsevier B.V 01.06.2008; Elsevier
• Subjects: Combinatorics; Combinatorics. Ordered structures; Compensated compactness; Designs and configurations; Error estimates; Exact sciences and technology; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Sciences and techniques of general use; Superconductors; Time discretization; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; 65M15; Superconductors; Error estimates; Compensated compactness; 82D55; Time discretization; 65M12; Approximation; Initial condition; Error estimation; Evolution equation; Numerical method; Boundary condition; Discretization method; Convergence; Non linear equation; Numerical analysis; Electric field; Applied mathematics; 65M12;65M15;82D55
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2006.03.055

In this paper we study a nonlinear evolution equation ∂ t ( σ ( | E | ) E ) + ∇ × ∇ × E = F in a bounded domain subject to appropriate initial and boundary conditions. This governs the evolution of the electric field E in a conductive medium under the influence of a force F . It is an approximation of Bean's critical-state model for type-II superconductors. We design a nonlinear numerical scheme for the time discretization. We prove the convergence of the proposed method. The proof is based on a generalization of div– curl lemma for transient problems. We also derive some error estimates for the approximate solution.