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A priori error analysis of residual-free bubbles for advection-diffusion problems. (English) Zbl 0947.65115

The subject of this work is that of developing stable and accurate finite element approximations to the steady linear advection-diffusion problem \[ \begin{aligned} -\varepsilon \nabla u + {\mathbf a}\cdot\nabla u = & f\;\;\text{ in} \Omega , \\ u = & 0\;\;\text{ on} \partial \Omega. \end{aligned} \] Here \(\Omega\) is a bounded polygonal domain in the plane, with boundary \(\partial \Omega\), \(\varepsilon\) is a positive constant, and \({\mathbf a}\) is a constant vector. Stable methods for this class of problems are generally characterised by a modification of the standard Galerkin method in which terms depending on the residual are added. A key example is the streamline-upwind Petrov-Galerkin method of A. N. Brooks and T. J. R. Hughes [Comput. Methods Appl. Mech. Eng. 32, 199-259 (1982; Zbl 0497.76041)]. A later development has been that of residual-free bubbles [cf. C. Baincchi, F. Brezzi and L. P. France, ibid. 105, No. 1, 125-141 (1993; Zbl 0772.76033)], in which the standard Galerkin space is enhanced by the addition of specially chosen bubble functions on each element.
In this work the authors undertake an analysis of the method of residual-free bubbles, for the case of piecewise linear continuous triangular elements. The bubble function is defined to be one that satisfies the partial differential equation strongly on the element, and which is zero on the element boundary. Properties of the bubble functions are established, after which the authors embark on a stability analysis. An optimal order error estimate is established in a “stability” norm \(\|\cdot \|_S\) defined by \[ \|v\|^2_S := \|\varepsilon \|\nabla v\|^2_{L^2} + \sum_T h_T \|{\mathbf a}\cdot\nabla v\|^2_{L^2}. \] The norm arises naturally in the analysis. This estimate is very similar to that obtained for the streamline diffusion method of C. Johnson, U. Nävert and J. Pitkäranta [ibid. 45, 285-312 (1984; Zbl 0537.76060)]. It is also worth noting that the estimate is for the entire solution, that is, the finite element approximation supplemented with residual-free bubbles.
The authors conclude their work by indicating how their analysis may be extended to problems in one and three dimensions, and to nonconstant data \(f\) and \({\mathbf a}\).

MSC:

65N15 Error bounds for boundary value problems involving PDEs
76R50 Diffusion
35F15 Boundary value problems for linear first-order PDEs
76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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