### Abstract

Language | English |
---|---|

Pages | 705-735 |

Number of pages | 30 |

Journal | numerical functional analysis and optimization |

Volume | 19 |

Issue number | 7-8 |

DOIs | |

Publication status | Published - 1998 |

### Fingerprint

### Keywords

- numerical analysis
- finite element
- dirichlet-to-neumann methods
- exterior transmission problems

### Cite this

*19*(7-8), 705-735. https://doi.org/10.1080/01630569808816855

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**On the numerical analysis of finite element and Dirichlet-to-Neumann methods for nonlinear exterior tramnsmission problems.** / Barrenechea, Gabriel R.; Barrientos, Mauricio A.; Gatica, Gabriel N.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the numerical analysis of finite element and Dirichlet-to-Neumann methods for nonlinear exterior tramnsmission problems

AU - Barrenechea, Gabriel R.

AU - Barrientos, Mauricio A.

AU - Gatica, Gabriel N.

PY - 1998

Y1 - 1998

N2 - We provide the numerical analysis of the combination of finite elements and Dirichlet-to-Neumann mappings (based on boundary integral operators) for a class of nonlinear exterior transmission problems whose weak formulations reduce to Lipschitz-continuous and strongly monotone operator equations. As a model we consider a nonlinear second order elliptic equation in divergence form in a bounded inner region of the plane, coupled with the Laplace equation in the corresponding unbounded exterior part. A discrete Galerkin scheme is presented by using linear finite elements on a triangulation of the domain, and then applying numerical quadrature and analytical formulae to evaluate all the linear, bilinear and semilinear forms involved. We prove the unique solvability of the discrete equations, and show the strong convergence of the approximate solutions. Furthermore, assuming additional regularity on the solution of the continuous operator equation, the asymptotic rate of convergence O(h) is also derived. Finally, numerical experiments are presented, which confirm the convergence results.

AB - We provide the numerical analysis of the combination of finite elements and Dirichlet-to-Neumann mappings (based on boundary integral operators) for a class of nonlinear exterior transmission problems whose weak formulations reduce to Lipschitz-continuous and strongly monotone operator equations. As a model we consider a nonlinear second order elliptic equation in divergence form in a bounded inner region of the plane, coupled with the Laplace equation in the corresponding unbounded exterior part. A discrete Galerkin scheme is presented by using linear finite elements on a triangulation of the domain, and then applying numerical quadrature and analytical formulae to evaluate all the linear, bilinear and semilinear forms involved. We prove the unique solvability of the discrete equations, and show the strong convergence of the approximate solutions. Furthermore, assuming additional regularity on the solution of the continuous operator equation, the asymptotic rate of convergence O(h) is also derived. Finally, numerical experiments are presented, which confirm the convergence results.

KW - numerical analysis

KW - finite element

KW - dirichlet-to-neumann methods

KW - exterior transmission problems

U2 - 10.1080/01630569808816855

DO - 10.1080/01630569808816855

M3 - Article

VL - 19

SP - 705

EP - 735

IS - 7-8

ER -