MATHEMATICS OF COMPUTATION Volume 72, Number 241, Pages 17–40 S 0025-5718(01)01378-3 Article electronically published on July 13, 2001 LAVRENTIEV REGULARIZATION + RITZ APPROXIMATION = UNIFORM FINITE ELEMENT ERROR ESTIMATES FOR DIFFERENTIAL EQUATIONS WITH ROUGH COEFFICIENTS ANDREW KNYAZEV AND OLOF WIDLUND Abstract. We consider a parametric family of boundary value problems for a diffusion equation with a diffusion coefficient equal to a small constant in a sub- domain. Such problems are not uniformly well-posed when the constant gets small. However, in a series of papers, Bakhvalov and Knyazev have suggested a natural splitting of the problem into two well-posed problems. Using this idea, we prove a uniform finite element error estimate for our model problem in the standard parameter-independent Sobolev norm. We also study uniform reg- ularity of the transmission problem, needed for approximation. A traditional finite element method with only one additional assumption, namely, that the boundary of the subdomain with the small coefficient does not cut any finite element, is considered. One interpretation of our main theorem is in terms of regularization. Our FEM problem can be viewed as resulting from a Lavrentiev regularization and a Ritz–Galerkin approximation of a symmetric ill-posed problem. Our error estimate can then be used to find an optimal regularization parameter together with the optimal dimension of the approximation subspace. 1. Introduction A particularly challenging class of problems arises with models described by partial differential equations (PDE’s) with highly discontinuous coefficients. Many important physical problems are of this nature. In particular, they arise in the design and study of composite materials built from essentially different components; see, e.g., [9, 38, 7, 34, 24]. The fictitious domain/embedding method is another source of PDE’s with highly discontinuous coefficients; cf., e.g., [39, 1, 12, 31]. In this method, the domain of the original boundary value problem is embedded into a larger one, where a new artificial boundary value problem is constructed. In the new, fictitious part of the domain the coefficients of PDE are chosen to be close to zero, if the original boundary condition is of Neumann type, or very large, in the Dirichlet case. Received by the editor May 19, 1998 and, in revised form, December 28, 2000. 2000 Mathematics Subject Classification. Primary 65N30, 35R05; Secondary 35J25, 35J70. Key words and phrases. Galerkin, Lavrentiev, Ritz, Tikhonov, discontinuous coefficients, er- ror estimate, finite elements, regularization, regularity, transmission problem, fictitious domain, embedding. The first author was supported by NSF Grant DMS-9501507. The second author was supported in part by NSF Grant CCR-9732208 and in part by the U.S. Department of Energy under contract DE-FG02-92ER25127. c 2001 American Mathematical Society 17 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use