Title: Least-squares approximation of div-curl systems


                            Joseph Pasciak
                      507B Blocker Hall, MS-3368
                         Texas A&M University
                    College Station, Tx 77843-3368
                     Email: pasciak@math.tamu.edu
              Web page: http://www.math.tamu.edu/~pasciak


Abstract:
In this talk, I will describe an approximation technique for div-curl systems 
based in (L2(D))3 where D is a domain in R3. Div-curl systems arise, for 
example, in electromagnetic applications. These systems are troublesome as 
they appear to be not elliptic and a crude counting indicates that there are 
more equations than unknowns. We formulate this problem as a general 
variational problem with different test and trial spaces. The analysis 
requires the verification of an appropriate inf-sup condition. This results 
is a very weak formulation where the solution space is (L2(D))3 and the data 
resides in various negative norm spaces. The advantages of setting up the 
problem in such a weak space will be discussed.

Subsequently, we consider finite element approximations based on this weak 
formulation. We present two possible approaches. The first involves the 
development of "stable pairs" of discrete test and trial spaces. With this 
approach, we enlarge the test space so that the discrete inf-sup condition 
holds and use a least-squares formulation to reduce to a uniquely solvable 
linear system. The second approach uses a smaller test space and adds terms 
to the form to stabilize the method. Both methods lead to optimal order 
estimates for problems with minimal regularity. This is important as it is 
easy to construct magnetostatic field applications whose solutions have low 
Sobolev regularity (e.g., (Hs(D))3 with 0< s< 1/2).