First-Order System Least-Squares for Second-Order Elliptic
Problems with Discontinuous Coefficients
Thomas A. Manteuffel
Stephen F. McCormick
Gerhard Starke
Abstract
The first-order system least-squares methodology represents an alternative to
standard mixed finite element methods. Among its advantages is the fact that
the finite element spaces approximating the pressure and flux variables are
not restricted by the inf-sup condition and that the least-squares functional
itself serves as an appropiate error measure. This paper studies the
first-order system least-squares approach for scalar second-order elliptic
boundary value problems with discontinuous coefficients. Ellipticity of an
appropriately scaled least-squares bilinear form is shown independently of the
size of the jumps in the coefficients leading to adequate finite element
approximation results. The occurrence of singularities at interface corners
and crosspoints is discussed, and a weighted least-squares functional is
introduced to handle such cases. Numerical experiments are presented for two
test problems to illustrate the performance of this approach.