## 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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