## Approximate symmetrization and Petrov-Galerkin methods for diffusion- convection problems.(English)Zbl 0562.76086

The authors study the following diffusion-convection problem: $$-\nabla \cdot (a\nabla u-bu)=f$$ in $$\Omega \in {\mathbb{R}}^ d$$, $$(a>0$$, $$\nabla \cdot b=0)$$; $$u=g$$ on $$\Gamma_ D$$, $$\partial u/\partial n=0$$ on $$\Gamma_ N$$, where the boundary $$\Gamma_ D\cup \Gamma_ N$$ of $$\Omega$$ is such that the Dirichlet section $$\Gamma_ D\neq \{\emptyset \}$$ includes all the inflow boundary, i.e., $$b\cdot n\geq 0$$ on $$\Gamma_ N$$, where $$n=outward$$ normal. By carefully introducing the concept of symmetrization, the authors derive general error bounds for a Petrov- Galerkin method, provide approximations based on the symmetric forms associated with the problem, they also address the problem of how to interpret a finite element approximation which attempts to achieve optimality in one of the symmetric forms and finally generate numerical examples for one- and two-dimensional cases.
Reviewer: M.A.Ibiejugba

### MSC:

 76Rxx Diffusion and convection 76M99 Basic methods in fluid mechanics
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### References:

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