### Abstract

The first-order div least squares finite element methods (LSFEMs) allow for an immediate a posteriori error control by the computable residual of the least squares functional. This paper establishes an adaptive refinement strategy based on some equivalent refinement indicators. Since the first-order div LSFEM measures the flux errors in H (div), the data resolution error measures the L ^{2} norm of the right-hand side f minus the piecewise polynomial approximation II f without a mesh-size factor. Hence the data resolution term is neither an oscillation nor of higher order and consequently requires a particular treatment, e.g., by the thresholding second algorithm due to Binev and DeVore. The resulting novel adaptive LSFEM with separate marking converges with optimal rates relative to the notion of a nonlinear approximation class.

Original language | English |
---|---|

Pages (from-to) | 43-62 |

Number of pages | 20 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 53 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2014 Jan 1 |

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### All Science Journal Classification (ASJC) codes

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

*SIAM Journal on Numerical Analysis*,

*53*(1), 43-62. https://doi.org/10.1137/130949634

}

*SIAM Journal on Numerical Analysis*, vol. 53, no. 1, pp. 43-62. https://doi.org/10.1137/130949634

**Convergence and optimality of adaptive least squares finite element methods.** / Carstensen, Carsten; Park, Eun-Jae.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Convergence and optimality of adaptive least squares finite element methods

AU - Carstensen, Carsten

AU - Park, Eun-Jae

PY - 2014/1/1

Y1 - 2014/1/1

N2 - The first-order div least squares finite element methods (LSFEMs) allow for an immediate a posteriori error control by the computable residual of the least squares functional. This paper establishes an adaptive refinement strategy based on some equivalent refinement indicators. Since the first-order div LSFEM measures the flux errors in H (div), the data resolution error measures the L 2 norm of the right-hand side f minus the piecewise polynomial approximation II f without a mesh-size factor. Hence the data resolution term is neither an oscillation nor of higher order and consequently requires a particular treatment, e.g., by the thresholding second algorithm due to Binev and DeVore. The resulting novel adaptive LSFEM with separate marking converges with optimal rates relative to the notion of a nonlinear approximation class.

AB - The first-order div least squares finite element methods (LSFEMs) allow for an immediate a posteriori error control by the computable residual of the least squares functional. This paper establishes an adaptive refinement strategy based on some equivalent refinement indicators. Since the first-order div LSFEM measures the flux errors in H (div), the data resolution error measures the L 2 norm of the right-hand side f minus the piecewise polynomial approximation II f without a mesh-size factor. Hence the data resolution term is neither an oscillation nor of higher order and consequently requires a particular treatment, e.g., by the thresholding second algorithm due to Binev and DeVore. The resulting novel adaptive LSFEM with separate marking converges with optimal rates relative to the notion of a nonlinear approximation class.

UR - http://www.scopus.com/inward/record.url?scp=84923943873&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84923943873&partnerID=8YFLogxK

U2 - 10.1137/130949634

DO - 10.1137/130949634

M3 - Article

AN - SCOPUS:84923943873

VL - 53

SP - 43

EP - 62

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 1

ER -