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

Abstract: We consider a parametric family of boundary value problems for the diffusion equation with the diffusion coefficient eual to a small constant in a subdomain. 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 consider 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. 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.

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