Numerical solution of nonlinear differential boundary value problems using adaptive non-overlapping domain decomposition method

This work aims is to study a nonlinear second-order boundary value differential elliptic problem in one dimension where the nonlinearity concerns the solution and its first derivative. We assume that the source term can be non-smooth and the nonlinearity can grow faster than quadratic. First, we sho...

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Vydáno v:Applicable analysis Ročník 101; číslo 6; s. 2044 - 2065
Hlavní autoři: Naceur, Nahed, Khenissi, Moez, Roche, Jean R.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Abingdon Taylor & Francis 13.04.2022
Taylor & Francis Ltd
Témata:
Algorithms
Analysis of PDEs
Approximation
Boundary value problems
Decomposition
domain decomposition
Domain decomposition methods
finite elements
Iterative methods
Mathematics
Newton methods
Nonlinear equations
Nonlinear ODE's
nonlinear PDE's
Nonlinearity
Numerical Analysis
Numerical methods
variational method
nonlinear PDE's
Nonlinear ODE's
variational method
domain decomposition
finite elements
ISSN:0003-6811, 1563-504X
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Abstract This work aims is to study a nonlinear second-order boundary value differential elliptic problem in one dimension where the nonlinearity concerns the solution and its first derivative. We assume that the source term can be non-smooth and the nonlinearity can grow faster than quadratic. First, we show the existence of a non-negative weak solution if we assume the existence of a super-solution. Second, we present a numerical algorithm to compute an approximation of the non-negative weak solution. The proposed algorithm is decomposed in two steps, the first one is devoted to computing a super-solution, and in the second one, the algorithm computes a sequence of solutions of an intermediate problem obtained by using the Yosida approximation of the nonlinearity. This sequence converges to the non-negative weak solution of the nonlinear equation. The numerical method is an application of the Newton method to the discretized version of the problem, but at each iteration, the resulting system can be indefinite. To overcome this difficulty, we introduce an adaptive non-overlapping domain decomposition method.
AbstractList This work aims is to study a nonlinear second-order boundary value differential elliptic problem in one dimension where the nonlinearity concerns the solution and its first derivative. We assume that the source term can be non-smooth and the nonlinearity can grow faster than quadratic. First, we show the existence of a non-negative weak solution if we assume the existence of a super-solution. Second, we present a numerical algorithm to compute an approximation of the non-negative weak solution. The proposed algorithm is decomposed in two steps, the first one is devoted to computing a super-solution, and in the second one, the algorithm computes a sequence of solutions of an intermediate problem obtained by using the Yosida approximation of the nonlinearity. This sequence converges to the non-negative weak solution of the nonlinear equation. The numerical method is an application of the Newton method to the discretized version of the problem, but at each iteration, the resulting system can be indefinite. To overcome this difficulty, we introduce an adaptive non-overlapping domain decomposition method.
Author Roche, Jean R.
Khenissi, Moez
Naceur, Nahed
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  givenname: Jean R.
  surname: Roche
  fullname: Roche, Jean R.
  organization: Université de Lorraine, CNRS, IECL
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Keywords nonlinear PDE's
Nonlinear ODE's
variational method
domain decomposition
finite elements
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Snippet This work aims is to study a nonlinear second-order boundary value differential elliptic problem in one dimension where the nonlinearity concerns the solution...
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SubjectTerms Algorithms
Analysis of PDEs
Approximation
Boundary value problems
Decomposition
domain decomposition
Domain decomposition methods
finite elements
Iterative methods
Mathematics
Newton methods
Nonlinear equations
Nonlinear ODE's
nonlinear PDE's
Nonlinearity
Numerical Analysis
Numerical methods
variational method
Title Numerical solution of nonlinear differential boundary value problems using adaptive non-overlapping domain decomposition method
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