Superconvergence analysis of two-grid FEM for Maxwell’s equations with a thermal effect. (English) Zbl 1446.65183
Summary: Based on two-grid algorithm, we develop the superconvergence analysis of fully-discrete scheme with the lowest-order Nédélec element and Crank-Nicolson scheme for the magneto-heat coupling model, which is also considered as Maxwell’s equations with a thermal effect. Then, main process of numerical analysis has two parts: On one hand, we utilize Newton-type Taylor expansion on superconvergent solutions for those nonlinear terms on coarse mesh, differing from the numerical solution, which makes our given two-grid method successful. On the other hand, we prove the solutions on the fine mesh to be higher accuracy by the postprocessing interpolation technique. Such a design is conducive to improving the computational accuracy in space and decreasing time consumption simultaneously. By employing the skill, we can obtain the convergent rate of \(O(\Delta t^2+h^2+H^3)\), which means that the space mesh size should satisfy \(h=O(H^{\frac{3}{2}})\). Finally, the numerical example is presented to verify the theoretical results.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |
| 78M20 | Finite difference methods applied to problems in optics and electromagnetic theory |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
| 65D05 | Numerical interpolation |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 78A25 | Electromagnetic theory (general) |
| 76V05 | Reaction effects in flows |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
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