number of iterationsrequired to meet the convergencecriterion. the converged solutions from the previous step. This significantly reduces the interfacial boundaries, the initial estimates for the interfacial flux is given from scheme. Outside of the first time step where zero initial flux is assumed on all between subdomains are satisfied using a Schwarz Neumann-Neumam iteration method which is...

An adjustment scheme for the regularization parameter of a Moreau-Yosida-based regularization, or relaxation, approach to the numerical solution of pointwise state constrained elliptic optimal control problems is introduced. The method utilizes error estimates of an associated finite element discretization of the regularized problems for the optimal selection of the regularization parameter in ...

Evolution Equations 254 Let F be another Hilbert space (with scalar product and 255 norm denoted by the same symbols as for H), the so256 called control space, and let B : F ! H be a linear opera257 tor, that we will assume to be bounded for the time being. 258 Then, given T > 0 and u0 2 H, for all f 2 L2(0; T ; F) the 259 Cauchy problem 260 ( u0(t) D Au(t)C B f (t) t 0 u(0) D u0 (12) 261 has a...

We encounter partial differential equations routinely in transport phenomena. Some examples are unsteady flow in a channel, steady heat transfer to a fluid flowing through a pipe, and mass transport to a falling liquid film. Here, we shall learn a method for solving partial differential equations that complements the technique of separation of variables. We shall also learn when the method can ...

In this article a boundary feedback stabilization approach for incompressible Navier-Stokes flow is studied. One of the main difficulties encountered is the fact that after space discretization by a mixed finite element method (because of the solenoidal condition) we end up with a differential-algebraic system (DAE) of index 2. The remedy here is to use a discrete realization of the Leray proje...

The midterm exam will be held on Thursday, March 25, 12-1:15, Krieger 304. Time for exam: 70 minutes. There will be 3 problem (100 points) plus 1 bonus problem (10 points). A sample exam is given along with the review, I will post the solution in these days. Remark. This review is intended as a guideline for this course. There may be some typos in this review. You should consult the textbook fo...

Differential constraints compatible with the linearized equations of partial differential equations are examined. Recursion operators are obtained by integrating the differential constraints. One of the standard ways for determining particular solutions to partial differential equations is to reduce them to ordinary differential equations which are easier to solve. The classical work of Lie abo...

In this paper we consider the optimal control problem governed by a time-dependent variational inequality arising in quasistatic plasticity with linear kinematic hardening. In the first part of the paper we address certain continuity properties of the forward operator, which implies the existence of an optimal control. Moreover, a discretization in time is derived and we show that every local m...