Abstract:In this paper we extend the ideas of the so-called validated continuation technique to the context of rigorously proving the existence of equilibria for partial differential equations defined on higherdimensional spatial domains. For that effect we present a new set of general analytic estimates. These estimates are valid for any dimension and are used, together with rigorous computations, to construct a finite number of radii polynomials. These polynomials provide a computationally efficient method to prove, … Show more
“…Now, there are some analytic convolution estimates (e.g. the ones developed in [11,13,15]) that allow computingZ M (r) satisfying (16). These explicit estimates essentially follow from the fact that the Banach space Ω q given in (4) is an algebra under discrete convolutions.…”
Section: The Radii Polynomial Approachmentioning
confidence: 99%
“…It can be shown that looking for solutions of (40) is equivalent to looking for solutions of f (x) = 0 in the Banach space X = R 2 × Ω q , where Ω q = {a = (a k ) k≥0 : a q < ∞} is the Banach space of infinite sequences algebraically decaying to 0 at least as fast as k −q with decay rate q > 1 (e.g. see [13]). The regularity estimates given by Lemma A.4 in Appendix A can be used to show that if a ∈ Ω q , then a 3 ∈ Ω q .…”
Section: Two-dimensional Manifold Of Equilibria Of Cahn-hilliardmentioning
confidence: 99%
“…We decided to present our method in the context of spectral Galerkin discretizations because the explicit analytic estimates required to control the truncation error terms (the tail part) involved in computing on a Galerkin projection have been recently developed within the field of rigorous numerics (e.g. see [11,13]). …”
In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite dimensional domain of the operator. The idea behind the construction of the smooth charts is to use the radii polynomial approach to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a twodimensional manifold of equilibria of the Cahn-Hilliard equation.
“…Now, there are some analytic convolution estimates (e.g. the ones developed in [11,13,15]) that allow computingZ M (r) satisfying (16). These explicit estimates essentially follow from the fact that the Banach space Ω q given in (4) is an algebra under discrete convolutions.…”
Section: The Radii Polynomial Approachmentioning
confidence: 99%
“…It can be shown that looking for solutions of (40) is equivalent to looking for solutions of f (x) = 0 in the Banach space X = R 2 × Ω q , where Ω q = {a = (a k ) k≥0 : a q < ∞} is the Banach space of infinite sequences algebraically decaying to 0 at least as fast as k −q with decay rate q > 1 (e.g. see [13]). The regularity estimates given by Lemma A.4 in Appendix A can be used to show that if a ∈ Ω q , then a 3 ∈ Ω q .…”
Section: Two-dimensional Manifold Of Equilibria Of Cahn-hilliardmentioning
confidence: 99%
“…We decided to present our method in the context of spectral Galerkin discretizations because the explicit analytic estimates required to control the truncation error terms (the tail part) involved in computing on a Galerkin projection have been recently developed within the field of rigorous numerics (e.g. see [11,13]). …”
In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite dimensional domain of the operator. The idea behind the construction of the smooth charts is to use the radii polynomial approach to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a twodimensional manifold of equilibria of the Cahn-Hilliard equation.
“…For instance, as mentioned in [13], "it is an interesting open problem to prove that both symmetric and asymmetric solutions coexist in S 1 ∩ A 2 ." The goal of the present paper is to prove these open questions for specific parameter values using the rigorous computational methods of [20][21][22][23][24] and more specifically with the approach as introduced in [25].…”
Section: (A))mentioning
confidence: 99%
“…All proofs can be found in [23,28]. Consider a decay rate s > 2, a computational parameter M > 6 and define, for k > 3,…”
Section: Coexistence Of Nontrivial Solutionsmentioning
In this paper, Chebyshev series and rigorous numerics are combined to compute solutions of the Euler-Lagrange equations for the one-dimensional Ginzburg-Landau model of superconductivity. The idea is to recast solutions as fixed points of a Newton-like operator defined on a Banach space of rapidly decaying Chebyshev coefficients. Analytic estimates, the radii polynomials and the contraction mapping theorem are combined to show existence of solutions near numerical approximations. Coexistence of as many as seven nontrivial solutions is proved.
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