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

Gabriel R. Barrenechea, Mauricio A. Barrientos, Gabriel N. Gatica

Research output: Contribution to journalArticle

Abstract

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.
LanguageEnglish
Pages705-735
Number of pages30
Journal numerical functional analysis and optimization
Volume19
Issue number7-8
DOIs
Publication statusPublished - 1998

Fingerprint

Exterior Problem
Operator Equation
Dirichlet
Nonlinear Problem
Numerical Analysis
Finite Element
Numerical Quadrature
Second Order Elliptic Equations
Weak Formulation
Transmission Problem
Unique Solvability
Monotone Operator
Boundary Integral
Discrete Equations
Laplace's equation
Strong Convergence
Semilinear
Integral Operator
Galerkin
Convergence Results

Keywords

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

Cite this

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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.

Vol. 19, No. 7-8, 1998, p. 705-735.

Research output: Contribution to journalArticle

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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

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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

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