Keywords: Bingham fluids, free boundary, non-linear, multivalued operator, finite elements, adaptive.

This paper concerns an adaptive finite element method for the flow of Bingham fluids. We study the steady flow through a cylinder under the effect of a pressure decay. This problem has been intensively studied for viscoelastic materials [3]. It may be seen as the starting point for the study of more complex flows. It can also be considered as a benchmark for testing the accuracy of the numerical method we present because it contains many difficulties of different kinds. The Bingham model [1,2] is characterized by the following property: the material starts to flow only if the applied forces exceed a certain limit , called the yield limit.The flow can be viewed as two kinds of distinct behaviors: viscous and rigid zones which leads to free boundary problem.

We design an adaptive algorithm based on a combination of the Uzawa method associated to the corresponding multivalued operator (procedure UPDATE) and on a convergent adaptive method for the linear problem (procedure ELLIPTIC). The idea of exploiting of our method is inspired by reference [4].

The procedure ELLIPTIC is based on a-posteriori residual estimate [5]. The general outline of the adaptive algorithm is the following

Elements where the error indicator is large will be marked for refinement, while elements with a small error indicator are left unchanged. We use the "Guaranteed error reduction strategy" to mark the elements [6]. The idea is to refine a subset of the triangulation whose element errors sum up to a fixed amount of the total error.

We use lower and upper bounds for a posteriori estimate to prove the convergence of the succession generated by cycle of the adaptive procedure.

The procedure UPDATE involves the equation of state of a Langrange multiplier. As it is well known, the Uzawa algorithm consists of solving for each iteration a linear problem and a nonlinear adaptation of the Lagrange multiplier associated with the multivalued operator. The existence of this multiplier is known [7], which assures the equivalence of the problem in terms of the multivalued operator and the variational formulation.

As our main result we show that, since the adaptive method for the linear problem is convergent, then, our adaptive modified Uzawa method is convergent as well.

As an application of the method described above, we model the laminar steady flow of a Bingham fluid in a cylindrical pipe. The results show the coherence with the theoretical analysis. We test our algorithm and compare with the classical Uzawa method. The results show a remarkable improvement of our method with respect to the classical method.

We justify the use of an a-posteriori error estimation for the linear problem that appears. Our main result is that our algorithm has the benefits of adaptivity to approach the solution with the minimum computational effort and also improves the convergence of the classical Uzawa method. We have tested the method and measure the good quality of the estimates and rates of convergence.

The numerical experiments have been developed with the finite element toolbox ALBERTA [8].

1

E. C. Bingham,"Fluidity and Plasticity", Mc. Graw-Hill, Nueva York, 1922.

2

J. G. Oldroyd,"A Rational Rormulation of the Equations of Plastic Flow for a Bingham Solid", Proc. Camb. Phil. Soc., 43, 100-105, 1947. doi:10.1017/S0305004100023239

3

P. Mosolov, V. P. Miasnikov, "Variational Methods in the Theory of the Fluidity of a Viscous Plastic Medium", P. M. M., 29 ,468-492, 1965.

4

E. Bänsch, P. Morin, R. H. Nochetto,An adaptive Uzawa FEM for the Stokes problem: convergence without the inf-sup condition, SIAM J. Numer. Anal., 40, no. 4, 1207-1229, 2002. doi:10.1137/S0036142901392134

5

R. Verfürth,A review of a posteriori error estimation and adaptive mesh-refinement techniques, Wiley-Teubner, 1995.

6

W. Döfler,A convergent adaptive algorithm for Poisson`s equation, SIAM J. Numer. Anal., 33, no. 3, 1106-1124, 1996. doi:10.1137/0733054

7

R. Glowinski, "Numerical Methods for Nonlinear Variational Problems", Springer-Verlag, New York, 1984.

8

A. Schmidt, K. G. Siebert, "Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA", Springer 06/2000, Berlín, 2005.

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