On the solution of the polynomial systems arising in the discretization of certain ODEs

Dratman, Ezequiel; Matera, Guillermo

doi:10.1007/s00607-009-0046-7

Cited by 7 publications

(16 citation statements)

References 28 publications

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“…On the other hand, Lemma 9 shows that ∥y ′′ − g(y)∥ < A. We conclude that y (3) is also uniformly bounded.…”

Section: Estimates For the Mesh-independence Principlementioning

confidence: 71%

“…Concerning the second assertion, we prove the Lipschitz continuity of each function arising in the expression of y (3) in (24). Recall that z is uniformly bounded by definition.…”

Section: Estimates For the Mesh-independence Principlementioning

confidence: 98%

“…In [7,3,4,8] we exhibited an algorithm which, for a given j ∈ N and ϵ ′ > 0, computes an ϵ ′ -approximation of the discrete system F j = 0, i.e., a point…”

Section: Computing An ϵ-Approximation Of the Solution Of (2)mentioning

confidence: 99%

“…For a given mesh size, the solutions of the discretization of (1) were studied in [7,3,4,8,9], for different conditions concerning g and f . In this paper we consider the behavior of the discretization of the solutions (1) as the mesh size tends to zero, that is, we aim to approximate the solutions of (1).…”

Section: Introductionmentioning

confidence: 99%

“…Such solutions arise as the stationary solutions of the semilinear heat equation with Neumann boundary conditions. In particular, the dynamics of the latter are usually described in terms of the solutions of the former (see, e.g., [1][2][3][4]). …”

Section: Introductionmentioning

confidence: 99%

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Newton’s method and a mesh-independence principle for certain semilinear boundary-value problems

Dratman

Matera

2016

Journal of Computational and Applied Mathematics

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MSC: 65L10 65L12 65H10 65H20 Keywords:Boundary-value problems Neumann boundary conditions Newton's method Newton-Mysovskikh's theorem Mesh-independence principle Complete cubic splines a b s t r a c tWe exhibit an algorithm which computes an ϵ-approximation of the positive solutions of a family of boundary-value problems with Neumann boundary conditions. Such solutions arise as the stationary solutions of a family of semilinear parabolic equations with Neumann boundary conditions. The algorithm is based on a finite-dimensional Newton iteration associated with a suitable discretized version of the problem under consideration. To determine the behavior of such a discrete iteration we establish an explicit mesh-independence principle. We apply a homotopy-continuation algorithm to compute a starting point of the discrete Newton iteration, and the discrete Newton iteration until an ϵ-approximation of the stationary solution is obtained. The algorithm performs roughly O((1/ϵ) 1/2 ) flops and function evaluations.

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“…On the other hand, Lemma 9 shows that ∥y ′′ − g(y)∥ < A. We conclude that y (3) is also uniformly bounded.…”

Section: Estimates For the Mesh-independence Principlementioning

confidence: 71%

“…Concerning the second assertion, we prove the Lipschitz continuity of each function arising in the expression of y (3) in (24). Recall that z is uniformly bounded by definition.…”

Section: Estimates For the Mesh-independence Principlementioning

confidence: 98%

“…In [7,3,4,8] we exhibited an algorithm which, for a given j ∈ N and ϵ ′ > 0, computes an ϵ ′ -approximation of the discrete system F j = 0, i.e., a point…”

Section: Computing An ϵ-Approximation Of the Solution Of (2)mentioning

confidence: 99%