Abstract
In this thesis, we focus on numerical analysis for several strongly coupled parabolic equation systems in physics and material. Although classical theory can handle some cases, usually it is hard to obtain optimal error estimates for coupled systems with more than one variable. The coupling may be so strong that accuracy of one component would be lower than other components with the same order approximation. Then, the low accurate variable will pollute other variables. In this case, classical theory requires using higher order approximation to the inaccurate variable. Sometimes, mesh sizes are also restricted.Two specific systems are time-dependent Ginzburg-Landau equations and equations describing incompressible miscible flow in porous media. Firstly, mixed formulation is applied into Ginzburg-Landau equations under temporal gauge. Because of the degeneracy of magnetic potential equations, numerical magnetic potential will always pollute numerical density of superconducting electron pairs. However, we can still have optimal error estimates based on a non-local projection, while for the same order approximation, classical analysis cannot lead to any convergence result. Another advantage of our scheme over previous scheme is that we have a preciser numerical magnetic field, which plays an important role in physics.
Then, we turn our glare to incompressible miscible flow in porous media. This is another deeply coupled system and numerical pressure will affect accuracy of numerical concentration through numerical Darcy velocity. Classical theory shows that it is necessary to use one order higher approximation to pressure and/or velocity compared with decoupled system. The main distribution is that we propose a new elliptic quasi-projection, which can overcome the difficulty. This elliptic quasi-projection can be helpful for analysis of different numerical methods, for example, combination of backward Euler and Galerkin methods or mixed methods, and it can deal with characteristic type approach in time direction. Furthermore, the elliptic quasi-projection can be extended to other similar deeply coupled system, such as nonlinear thermistor equations. We would like to emphasize that all error estimates we obtain are unconditional. Our numerical schemes are efficient in practice, and we have performed a lot of numerical tests.
| Date of Award | 27 Aug 2020 |
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| Original language | English |
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| Supervisor | Weifeng QIU (Supervisor) & Weiwei Sun (External Co-Supervisor) |