By Gilbert Strang (auth.), David Y. Gao, Hanif D. Sherali (eds.)

The articles that include this exclusive annual quantity for the *Advances in Mechanics and Mathematics* sequence were written in honor of Gilbert Strang, a global popular mathematician and unprecedented individual. Written through best specialists in complementarity, duality, worldwide optimization, and quantum computations, this assortment finds the wonderful thing about those mathematical disciplines and investigates contemporary advancements in international optimization, nonconvex and nonsmooth research, nonlinear programming, theoretical and engineering mechanics, huge scale computation, quantum algorithms and computation, and data theory.

Much of the cloth, together with many of the methodologies, is written for nonexperts and is meant to stimulate graduate scholars and younger college to enterprise into this wealthy area of study; it's going to additionally profit researchers and practitioners in different parts of utilized arithmetic, mechanics, and engineering.

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Nozawa, Max-flow min-cut theorem in an anisotropic network, Osaka J. Math. 27 (1990) 805—842. 24. M. L. Overton, Numerical solution of a model problem from collapse load analysis, Computing Methods in Applied Science and Engineering VI, R. Glowinski and J. L. , Elsevier, 1984. 25. S. T. Rachev and L. R¨ uschendorf, Mass Transportation Problems I, II, Springer (1998). 26. S. Roy and I. J. Cox, A maximum-flow formulation of the n-camera stereo correspondence problem, Proc. Int. Conf. Computer Vision (1988) 492—499.

2 Model Elliptic Variational Inequality and Its Finite Element Approximation In this section, we introduce a model elliptic variational inequality of the second kind. 7 we provide a posteriori error analysis for the finite element solution of a steady-state frictional contact problem. Let Ω be a domain in Rd , d ≥ 1, with a Lipschitz boundary Γ . Let Γ1 ⊂ Γ be a relatively closed subset of Γ , and denote Γ2 = Γ \Γ1 the remaining part of the boundary. , Γ2 = ∅). e. on Γ . e. on Γ . Assume f ∈ L2 (Ω) and g > 0 are given.

10) , a ∈ Nv,0 . e a ϕa dx K Then define the interpolation operator Πh : V → Vh as follows: X Πh v = va ϕa . 11) a ∈Nv,0 The next result summarizes some basic estimates for Πh . Its proof can be found in [21]. 2. There exists an h-independent constant C > 0 such that for all v ∈ V and f ∈ L2 (Ω), 34 V. Bostan, W. 14) kh−1/2 (v − Πh v)k20;γ ≤ C|v|21;Ω . 5) is uh ∈ Vh , a(uh , vh −uh )+j(vh )−j(uh ) ≥ (vh −uh ) ∀ vh ∈ Vh . 16) The discrete problem has a unique solution uh ∈ Vh by the standard existence and uniqueness result on elliptic variational inequalities.