# zbMATH — the first resource for mathematics

A generalized duality method for solving variational inequalities. Applications to some nonlinear Dirichlet problems. (English) Zbl 1085.65052
The authors extend the results of C. Parés, J. Macías, and M. Castro [ibid. 89, No. 1, 161–189 (2001; Zbl 0991.65057)] to some problems where the definition of the functional $$j(v)= \Phi(\eta_0+ Bv)$$ involves the $$L^p(\Omega)^M$$ spaces. In this case, applying a similar reasoning, the optimal choice of $$\lambda(x)$$ and $$\omega(x)$$ should be given by two matrix-valued functions, as they are related to the Hessian of the convex functions involved.
Main result: The authors present a generalization of the algorithm allowing this choice of matrix-valued parameters. An abstract algorithm in a more general context where $$\lambda$$ and $$\omega$$ are elements of two classes of sufficiently general linear operators, so that constants, scalar or matrix-valued functions can be considered as particular cases. The generalized Bermúdez-Moreno algorithm is defined [cf. A. Bermúdez and C. Moreno, Comput. Math. Appl. 7, 43–58 (1981; Zbl 0456.65036)] and theorems of convergence are proved. The expression for the optimal values of $$\lambda$$ and $$\omega$$ for regular general problems are discussed. The optimal values depend on the exact solution, so that some variants are presented, based on an initial guess and periodic recalculations if needed.
Finally, the three proposed algorithms are applied to some boundary value problems related to the $$p$$-Laplacian operator with an exact known solution. Additionally, a faster algorithm for calculating Yosida regularizations with matrix-valued parameters is also presented.

##### MSC:
 65K10 Numerical optimization and variational techniques 49J40 Variational inequalities
Full Text:
##### References:
 [1] Andrews, K.T., Chapman, L., Fernández J.R. et al.: A membrane in adhesive contact. SIAM J. App. Math., 64(1), 152-169 (2003) · Zbl 1081.74034 [2] Bermúdez, A., Moreno, C.: Duality methods for solving variational inequalities. Comp. and Maths. with Appls., 7, 43-58 (1981) · Zbl 0456.65036 [3] Brézis, H.: Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Banach. North-Holland, 1973 · Zbl 0252.47055 [4] Calvo, N., Durany, J., Vázquez, C.: Finite elements numerical solution of a coupled profile-velocity-temperature shallow ice sheet approximation model. J. Comp. App. Math., 158(1), 31-41 (2003) · Zbl 1107.86300 [5] Calvo, N., Díaz, J.I., Durany, J., Schiavi, E., Vázquez, C.: On a doubly nonlinear parabolic obstacle problem modelling ice sheet dynamics. SIAM J. App. Math., 63(2), 683-707 (2003) · Zbl 1032.35115 [6] Duvaut, G., Lions, J.L.: Les Inéquations en Mécanique et en Physique. Dunod, 1972 · Zbl 0298.73001 [7] Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. volume 1 of Studies in Mathematics and its Applications, North-Holland, 1976 · Zbl 0322.90046 [8] Fortin, M., Glowinski, R.: Augmented Lagrangians: Applications to the Numerical Solution of Boundary Value Problems. North-Holland, 1983 · Zbl 0525.65045 [9] Glowinski, R., LeTallec, P.: Augmented Lagrangians and Operator-Splitting Methods in Nonlinear Mechanics. SIAM, 1989 [10] Krasnosel?skii, M.A.: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon Press, 1964 [11] Lions, J.L.: Quelques méthodes de resolution de problémes aux limites non linéaires. Dunod, Paris, 1969 [12] Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Num. Anal., 16, 964-979 (1979) · Zbl 0426.65050 [13] Parés, C., Macías, J., Castro, M.: Duality methods with an automatic choice of parameters. Numer. Math., 89, 161-189 (2001) · Zbl 0991.65057 [14] Parés, C., Castro, M., Macías, J.: On the convergence of the Bermúdez-Moreno algorithm with constant parameters. Numer. Math. 92, 113-128 (2002) · Zbl 1003.65069 [15] Pazy, A.: On the asymptotic behavior of iterates of nonexpansive mappings in Hilbert spaces. Israel J. Math., 26, 197-204 (1977) · Zbl 0343.47047 [16] Smagorinsky, J.: General circulation experiments with the primitive equations. Mon. Wea. Rev., 91(3), 99-165 (1963)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.