### 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 language | English (US) |
---|---|

Pages (from-to) | 17-40 |

Number of pages | 24 |

Journal | Mathematics of Computation |

Volume | 71 |

Issue number | 241 |

State | Published - Jan 2003 |

### Fingerprint

### 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

*Mathematics of Computation*,

*71*(241), 17-40.

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

Research output: Contribution to journal › Article

*Mathematics of Computation*, vol. 71, no. 241, pp. 17-40.

}

TY - JOUR

T1 - Lavrentiev regularisation + Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients

AU - Knyazev, Andrew

AU - Widlund, Olof

PY - 2003/1

Y1 - 2003/1

N2 - 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.

AB - 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.

KW - Discontinuous coefficients

KW - Embedding

KW - Error estimate

KW - Fictitious domain

KW - Finite elements

KW - Galerkin

KW - Lavrentiev

KW - Regularity

KW - Regularization

KW - Ritz

KW - Tikhonov

KW - Transmission problem

UR - http://www.scopus.com/inward/record.url?scp=0037236533&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037236533&partnerID=8YFLogxK

M3 - Article

VL - 71

SP - 17

EP - 40

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 241

ER -