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Analysis of least-squares mixed finite element methods for nonlinear nonstationary convection-diffusion problems. (English) Zbl 0947.65108
Nonlinear nonstationary convection-diffusion problems are discussed, especially from the numerical point of view. A least squares finite element method for solving this nonlinear convection-diffusion problem in a first-order system form is used. A new modified scheme is formulated in the sense of a weighted ${L}_{2}$ norm. A new optimal a priori error estimate in ${L}_{2}$ norm is proved if the classical mixed elements are used. Four fully discrete least squares mixed finite element schemes for the nonlinear first-order system are formulated and systematic theories on convergence of these schemes are established.
##### MSC:
 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) 35K55 Nonlinear parabolic equations 65M12 Stability and convergence of numerical methods (IVP of PDE) 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 76M10 Finite element methods (fluid mechanics) 76R05 Forced convection (fluid mechanics)