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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
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