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

Hoppe, R. H. W.; Kornhuber, R.

SIAM J. Numer. Anal. 31, No. 2, 301-323 (1994).

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.

Reviewer: T.Roubíček (Praha)

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

Keywords:

second-order elliptic variational inequality; finite elements; relaxation method; convergence; multilevel techniques; preconditioners; error estimates; numerical experiments

Software:

PLTMG