  We consider linear systems arising from the use of the finite element method
for solving scalar linear elliptic problems. Our main result is that these
linear systems, which are symmetric and positive semidefinite, are well
approximated by symmetric diagonally dominant matrices. Our framework for
defining matrix approximation is support theory. Significant graph theoretic
work has already been developed in the support framework for preconditioners in
the diagonally dominant case, and in particular it is known that such systems
can be solved with iterative methods in nearly linear time. Thus, our
approximation result implies that these graph theoretic techniques can also
solve a class of finite element problems in nearly linear time. We show that
the support number bounds, which control the number of iterations in the
preconditioned iterative solver, depend on mesh quality measures but not on the
problem size or shape of the domain.
