Stable approximations of a minimal surface problem with variational inequalities. (1997) - Nashed, M. Zuhair; Scherzer, Otmar
Abstract : In this paper we develop a new approach for the stable approximation of a minimal surface problem associated with a relaxed Dirichlet problem in the spaceB V ( Ω ) of functions of bounded variation. The problem can be reformulated as an unconstrained minimization problem of a functional𝒥 onB V ( Ω ) defined by𝒥 ( u ) = 𝒜 ( u ) + ∫ ∂ Ω | T u − Φ |, where𝒜 ( u ) is the "area integral" ofu with respect toΩ, T is the "trace operator" fromB V ( Ω ) intoL i ( ∂ Ω ), andϕ is the prescribed data on the boundary ofΩ . We establish convergence and stability of approximate regularized solutions which are solutions of a family of variational inequalities. We also prove convergence of an iterative method based on Uzawa's algorithm for implementation of our regularization procedure.