SUMMARYIn this paper, we apply asymptotic-numerical methods for computing non-linear equilibrium paths of elastic beam, plate and shell structures. The non-linear branches are sought in the form of asymptotic expansions, and they are determined by solving numerically (FEM) several linear problems with a single stiffness matrix.A large number of terms of the series can be easily computed by using recurrence formulas. In comparison with a more classical step-by-step procedure, the method is rapid and automatic. We show, with some examples, that the choice of the expansion's parameter and the use of Pade approximants play an important role in the determination of the size of the domain of convergence.

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SUMMARYNew predictor-corrector algorithms are presented for the computation of solution paths of non-linear partial di erential equations. The predictors and the correctors are based on perturbation techniques and Padà e approximants. This extends the Asymptotic Numerical Method (ANM), which is an e cient highorder continuation technique without corrector. The e ciency and the reliability of the new technique are assessed by several examples within thin shell theory and Navier-Stokes equations. Many variants have been tested to establish an optimal algorithm.

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Summary In this paper, we propose for the first time to extend the application field of the high‐order mesh‐free approach to the stationary incompressible Navier‐Stokes equations. This approach is based on a high‐order algorithm, which combines a Taylor series expansion, a continuation technique, and a moving least squares (MLS) method. The Taylor series expansion permits to transform the nonlinear problem into a succession of continuous linear ones with the same tangent operator. The MLS method is used to transform the succession of continuous linear problems into discrete ones. The continuation technique allows to compute step‐by‐step the whole solution of the discrete problems. This mesh‐free approach is tested on three examples: a flow around a cylindrical obstacle, a flow in a sudden expansion, and the standard benchmark lid‐driven cavity flow. A comparison of the obtained results with those computed by the Newton‐Raphson method with MLS, the high‐order continuation with finite element method, and those of literature is presented.

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