Lavrentiev regularisation + Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients

Andrew Knyazev, Olof Widlund

Research output: Contribution to journalArticle

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 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 also study uniform regularity 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.

Original languageEnglish (US)
Pages (from-to)17-40
Number of pages24
JournalMathematics of Computation
Volume71
Issue number241
StatePublished - Jan 2003

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Uniform Approximation
Rough
Error Estimates
Regularization
Differential equations
Finite Element
Differential equation
Finite element method
Coefficient
Boundary value problems
Well-posed Problem
Transmission Problem
Galerkin Approximation
Regularization Parameter
Optimal Parameter
Ill-posed Problem
Approximation
Diffusion equation
Diffusion Coefficient
Finite Element Method

Keywords

  • Discontinuous coefficients
  • Embedding
  • Error estimate
  • Fictitious domain
  • Finite elements
  • Galerkin
  • Lavrentiev
  • Regularity
  • Regularization
  • Ritz
  • Tikhonov
  • Transmission problem

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Applied Mathematics
  • Computational Mathematics

Cite this

Lavrentiev regularisation + Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients. / Knyazev, Andrew; Widlund, Olof.

In: Mathematics of Computation, Vol. 71, No. 241, 01.2003, p. 17-40.

Research output: Contribution to journalArticle

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