The University of Chicago Numerical Solution of Variational Inequalities a Dissertation Submitted to the Faculty of the Division of the Physical Sciences in Candidacy for the Degree of Doctor of Philosophy Department of Mathematics by Yongmin Zhang

If one wants to minimize a nonlinear functional it is often fruitful to consider the relationships which must hold at a minimum If the functional is di erentiable and the permitted variations at the minimum constitute a linear space this process gives equations that the minimum must satisfy and if the functional is quadratic these equations are linear However if the set of permitted variations is constrained for example to nonnegative functions or the functional is nondi erentiable then one may nd inequalities instead of equations We are interested in numerically approximating solutions of two types of variational inequalities The rst one is variational inequalities with constrained admissible set frequently called obstacle problems The second type is variational inequalities with a non di erentiable term An important example of this type is rigid visco plastic Bingham uid L error estimates for numerical solutions of obstacle problems have been in vestigated by C Baiocchi and J Nitsche Though Nitsche s estimate is optimal O h jlnhj the discrete solution he de ned is not in general computable because the obstacle it self is not discretized A new monotonicity principle for a discrete obstacle problems is applied to obtain an optimal L error estimate for an approximation in which the obstacle is only respected at the vertices of the triangulation This result both uses and improves the Nitsche s estimate Numerical computation of Bingham uid ow has been studied by M Fortin and R Glowinski but error estimates are not yet available for their methods A new numerical method for approximate solution of time dependent ow of Bingham uid in cylindrical pipes which uses regularization of nondi erentiable term is studied Error estimates are described for the case in which the discretization is done using piecewise linear nite elements in space and backward di erencing in time