Numerical solution of Variational Inequalities by Adaptive Finite Elements

Franz-Theo Suttmeier describes a general approach to a posteriori error estimation

and adaptive mesh design for finite element models where the solution

is subjected to inequality constraints. This is an extension to variational

inequalities of the so-called Dual-Weighted-Residual method (DWR method)

which is based on a variational formulation of the problem and uses global

duality arguments for deriving weighted a posteriori error estimates with respect

to arbitrary functionals of the error. In these estimates local residuals of

the computed solution are multiplied by sensitivity factors which are obtained

from a numerically computed dual solution. The resulting local error indicators

are used in a feed-back process for generating economical meshes which

are tailored according to the particular goal of the computation. This method

is developed here for several model problems. Based on these examples, a general

concept is proposed, which provides a systematic way of adaptive error

control for problems stated in form of variational inequalities.