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Stabilized finite element methods for singularly perturbed parabolic problems. (English) Zbl 0838.65095
The authors consider two stabilized numerical schemes for solving Dirichlet-type initial-boundary value problems of the parabolic equation $$(\partial_t + L_\varepsilon) u : = \partial_t u - \varepsilon \Delta u + b \cdot \nabla u + cu = f$$, modelling convection-reaction-diffusion. Weighted least squares forms of the underlying equation are added to the basic Galerkin finite element semidiscretization in order to accomodate the method to (possibly locally varying) convective, reactive or diffusive terms. Time integration is performed with standard stable one-step methods. Stability and error estimates are derived on unstructured grids and the design of the numerical damping parameters are considered. Numerical results are given for typical model problems, including “translating cone”.

##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 35K57 Reaction-diffusion equations 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 35B25 Singular perturbations in context of PDEs
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