Keywords: augmented Lagrangian, a-posteriori error estimators, adaptive mesh, convergence, multilevel preconditioners.

In this paper we present a new adaptive strategy to obtain an incompressible wind field that adjusts to an experimental one, and verify boundary conditions of physical interest [1].

We use an Augmented Lagrangian formulation for solving this problem (see [1]). Our method is based on Uzawa iteration to update the Lagrange multiplier on an elliptic adaptive inner iteration for velocity. For a positive diagonal matrix A, we derive a posteriori error estimator for the operator (inner iteration),

with an error reduction property, i.e. in each step of refinement we guarantee that the error decreases in a fixed proportion of the current error. Then, we extend the theory of [2,3] to this case, and prove that our algorithm converges within any prescribed tolerance in a finite number of steps without any preliminary mesh adaptation.

The numerical solution is approached with the lowest Raviart-Thomas element. Therefore, we have continuity of the flows through the inter-element boundaries, and the incompressibility contition at discrete level.

We solve the system of equations, corresponding to the inner loop, using a CG preconditioned by one V-cycle of Arnold-Falk-Winter multigrid [4]. Though the theoretical justification of this solver is only for quasiuniform triangulation, we obtain optimal results on adaptive meshes, i.e. the number of iterations remain bounded as the number of degree of freedom grows.

Several examples show that the proposed method is efficient and reliable. The numerical experiments have been designed with the finite element toolboox ALBERT [5], extended with new designed tools for the a-posteriori error estimator developed in this paper, and the multigrid preconditioner of [4].

Table 1: Results for obstacle model test. Notation: OUT/INT: th iteration of outer/inner loop. DOF: number of degrees of freedom for Raviart-Thomas element. LEVELS: number of multigrid levels. ITER: number of iterations for PCG. : error estimator for wind field. EOC: experimental orders of convergence (optimal order 1.0).

1

G. Winter, G. Montero, L. Ferragut and R. Montenegro. "Adaptive Strategies
Using Standard and Mixed Finite Elements for Wind Field Adjustment". Solar Energy, 54, 49-56, 1995. doi:10.1016/0038-092X(94)00100-R

2

M. Fortin and R. Glowinski, "Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary Value Problems". North Holland, Studies in Mathematics and its Applications, vol. 15, Amsterdam, 1983.

3

E. Bänsch, P. Morin, and R. H. Nochetto, "An Adaptive Uzawa FEM for the Stokes Problem: Convergence without the inf-sup Condition". SIAM J. Numer. Anal., 40, 1207-1229, 2002. doi:10.1137/S0036142901392134

4

P. Morin , R. H. Nochetto and K. G. Siebert, "Convergence of Adaptive Finite Element Methods". SIAM Review, 44, 631-658, 2002. doi:10.1137/S0036144502409093

5

D. N. Arnold, R. S. Falk and R. Winther, "Preconditioning in and Applications". Math. Comp., 66, 957-984, 1997. doi:10.1090/S0025-5718-97-00826-0

6

A. Schmidt and K. G. Siebert. "ALBERT: An Adaptive Hierarchical Finite Element Toolbox." Preprint 06/2000, Freiburg, 244 pp., 2000.

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