International audienceMany differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Lie-group structure under discretization is often vital in the recovery of qualitatively-correct geometry and dynamics and in the minimisation of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Lie-group structure, highlighting theory, algorithmic issues and a number of applications

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Many numerical algorithms involve computations in Lie algebras, like composition and splitting methods, methods involving the Baker-Campbell-Hausdorff formula and the recently developed Lie group methods for integration of differential equations on manifolds. This paper is concerned with complexity and optimization of such computations in the general case where the Lie algebra is free, i.e. no specific assumptions are made about its structure. It is shown how transformations applied to the original variables of a problem yield elements of a graded free Lie algebra whose homogeneous subspaces are of much smaller dimension than the original ungraded one. This can lead to substantial reduction of the number of commutator computations. Witt's formula for counting commutators in a free Lie algebra is generalized to the case of a general grading. This provides good bounds on the complexity. The interplay between symbolic and numerical computations is also discussed, exemplified by the new Matlab toolbox 'DiffMan'.

A commutative but not cocommutative graded Hopf algebra H N , based on ordered (planar) rooted trees, is studied. This Hopf algebra is a generalization of the Hopf algebraic structure of unordered rooted trees H C , developed by Butcher in his study of Runge-Kutta methods and later rediscovered by Connes and Moscovici in the context of noncommutative geometry and by Kreimer where it is used to describe renormalization in quantum field theory. It is shown that H N is naturally obtained from a universal object in a category of noncommutative derivations and, in particular, it forms a foundation for the study of numerical integrators based on noncommutative Lie group actions on a manifold. Recursive and nonrecursive definitions of the coproduct and the antipode are derived. The relationship between H N and four other Hopf algebras is discussed. The dual of H N is a Hopf algebra of Grossman and Larson based on ordered rooted trees. The Hopf algebra H C of Butcher, Connes, and Kreimer is identified as a proper Hopf subalgebra of H N using the image of a tree symmetrization operator. The Hopf algebraic structure of the shuffle algebra H Sh is obtained from H N by a quotient construction. The Hopf algebra H P of ordered trees by Foissy differs from H N in the definition of the product (noncommutative Date concatenation for H P and shuffle for H N ) and the definitions of the coproduct and the antipode, however, these are related through the tree symmetrization operator.

Appeared as section 5 in [17]. AbstractWe develop the theory of mixed finite elements in terms of special inverse systems of complexes of differential forms, defined over cellular complexes. Inclusion of cells corresponds to pullback of forms. The theory covers for instance composite piecewise polynomial finite elements of variable order over polyhedral grids. Under natural algebraic and metric conditions, interpolators and smoothers are constructed, which commute with the exterior derivative and whose product is uniformly stable in Lebesgue spaces. As a consequence we obtain not only eigenpair approximation for the Hodge-Laplacian in mixed form, but also variants of Sobolev injections and translation estimates adapted to variational discretizations.

Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on differential manifolds. They have been extensively studied in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic differential equations and renormalization. Butcher series are formal power series founded on pre-Lie algebras, used in numerical analysis to study geometric properties of flows on euclidean spaces. Motivated by the analysis of flows on manifolds and homogeneous spaces, we investigate algebras arising from flat connections with constant torsion, leading to the definition of post-Lie algebras, a generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately associated with euclidean geometry, post-Lie algebras occur naturally in the differential geometry of homogeneous spaces, and are also closely related to Cartan's method of moving frames. Lie-Butcher series combine Butcher series with Lie series and are used to analyze flows on manifolds. In this paper we show that Lie-Butcher series are founded on post-Lie algebras. The functorial relations between post-Lie algebras and their enveloping algebras, called D-algebras, are explored. Furthermore, we develop new formulas for computations in free post-Lie algebras and D-algebras, based on recursions in a magma, and we show that Lie-Butcher series are related to invariants of curves described by moving frames.

.Runge-Kutta methods are formulated via coordinate independent operations on manifolds . It is shown that there is an intimate connection between Lie series and Lie groups on one hand and Butcher's celebrated theory of order conditions on the other. In Butcher's theory the elementary differentials are represented as trees . In the present formulation they appear as commutators between vector fields . This leads to a theory for the order conditions, which can be developed in a completely coordinate free manner . Although this theory is developed in a language that is not widely used in applied mathematics, it is structurally simple . The recursion for the order conditions rests mainly on three lemmas, each with very short proofs . The techniques used in the analysis are prepared for studying RK-like methods on general Lie groups and homogeneous manifolds, but these themes are not studied in detail within the present paper .