# On the convergence to equilibria of a sequence defined by an implicite scheme

Abstract : We present numerical implicit scheme based on a geometric approach to the study of the convergence of solutions of gradient-like systems given in [2]. Depending on the globality of the induced metric, we can prove the convergence of these algorithms. Dedicated to the memory of Ezzeddine ZAHROUNI 1. Notation For a riemannian manifold (M, g) of dimension N we denote ·, · g the scalar product dened on each tangent space. The induced norm is denoted · g (or · when there is no risk of confusion) For a local system of coordinates on M , g ij will denote the coecient of the matrix dening the scalar product above. Let us recall that a C 1 curve x : [0, 1] → M is called a geodesic between x(0) and x(1) i it is a critical point of the functional L(γ) = 1 0 ||γ (t)|| g dt restricted to the C 1-curves γ : [0, 1] → M such that γ(0) = x(0) and γ(1) = x(1). For a dierentiable function f : M → R and p ∈ M we denote ∇ g f (p) the unique element of the tangent space T p M to M at p such that ∀u ∈ T p M, ∇ g f (p), u g = df (p).u 2. A implicit numerical scheme and main result of the paper Let us consider (M, g) a complete connected non compact riemaniann manifold and E a smooth real function. Associated to E, it is quite natural to consider the following gradient system (1)Ẋ(t) + ∇ g E(X(t)) = 0. In the paper [11] the authors Merlet & Pierre consider the situation when (M, g) is the standard R N with its natural euclidian structure and prove the convergence of a sequence dened by an implicit scheme associated to (1). It is quite natural to extend the scheme there introduced to the case of more general manifolds. Such insights were initially considered in [12] provided (M, g) is a submanifold of R N. However the specic case of the backward Euler scheme was not considered in this paper under the intrinsic point of view, i.e. the backward scheme is constructed ex post in [12], considering the embedded situation. Here we try to focus on the The rst author wishes to thanks the organizers of ICAAM 2019 in Hammamet, Tunisia, where this work was initiated. The second author wishes to thanks CNAM, France where this work was partially completed.
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https://hal-cnam.archives-ouvertes.fr/hal-02941522
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Submitted on : Thursday, September 17, 2020 - 10:08:54 AM
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Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicite scheme. 2020. ⟨hal-02941522⟩

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