The Obstacle Problem
The obstacle problem describes constrained deflection of a membrane and serves as a model problem for infinite-dimensional minimization problems with pointwise inequality constraint. Equivalent formulations of the problem including a variational inequality and a characterization with a Lagrange multiplier are derived. The well-posedness and convergence of lowest-order conforming finite element methods, together with a priori and a posteriori error estimates, are established. The application of the semismooth Newton method and a globally convergent primal-dual method together with their implementation conclude the chapter.
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