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

Barrett, J. W.; Morton, K. W.

Comput. Methods Appl. Mech. Eng. 45, 97-122 (1984).

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

Keywords:

diffusion-convection problem; symmetrization; error bounds; Petrov- Galerkin method; numerical examples

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