Lavrentiev Regularization + Ritz Approximation Uniform Finite Element
Error Estimates for Differential Equations with Rough Coefficients

Andrew Knyazev and Olof Widlund
May 1, 1998
URN = ncstrl.cudenver_ccm/UCD-CCM-132
http://cs-tr.cs.cornell.edu:80/Dienst/UI/1.0/Display/ncstrl.cudenver_ccm/UCD-CCM-132

Abstract:  We consider a parametric family of boundary va= lue problems for
the diffusion equation with the diffusion coefficient eq= ual to a small
constant in a subdomain.  Such problems are not uniformly w= ell-posed when
the constant gets small.  However, in a series of papers, B= akhvalov and
Knyazev have suggested a natural splitting of the problem in= to two
well-posed problems.  Using this idea, we prove a uniform finite el= ement
error estimate for our model problem in the standard parameter-inde= pendent
Sobolev norm.  We consider a traditional finite element method wit= h only one
additional assumption, namely, that the boundary of the subdom= ain with the
small coefficient does not cut any finite element.  One inter= pretation of
our main theorem is in terms of regularization.  Our FEM prob= lem can be
viewed as resulting from a Lavrentiev regularization and a Rit= z--Galerkin
approximation of a symmetric ill-posed problem.  Our error est= imate can then
be used to find an optimal regularization parameter togeth= er with the
optimal dimension of the approximation subspace.