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Adaptive multilevel methods for obstacle problems. (English) Zbl 0806.65064

On a two-dimensional polygonal domain \(\Omega\), the authors consider a second-order elliptic variational inequality of the type \(\int_ \Omega(\nabla u)^ T A(x)\nabla(u- v)\leq \int_ \Omega f(x)(u- v)\) for any \(v\) belonging to the Sobolev space \(H^ 1(\Omega)\) and satisfying the obstacle condition \(v\leq \varphi\) on \(\Omega\). The matrix \(A(x)\) is symmetric positive definite uniformly with respect to \(x\in \Omega\), which ensures existence and uniqueness of the solution \(u\in H^ 1(\Omega)\) satisfying \(v\leq \varphi\) on \(\Omega\). This problem is then discretized by piecewise linear finite elements, using a triangulation of \(\Omega\).
A projected relaxation method to solve the resulting discrete variational inequality can substantially improve its convergence by using multilevel techniques with respect to a hierarchy of triangulations together with a linearization technique, using at each iteration a prespecified set of active constraints. The authors construct and analyze multilevel preconditioners. Semilocal and local a posteriori error estimates are derived, and numerical experiments supporting the theoretical findings are presented, too.

MSC:

65K10 Numerical optimization and variational techniques
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
49M15 Newton-type methods
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
49J40 Variational inequalities
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs

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