Resumen de Padé / least-squares schemes for convective transport problems
Jean Donea 
, Bernardino Roig, Antonio Huerta Cerezuela
In this study we consider the development of high-order accurate numerical methods for solving purely convective transport problems without stability restrictions. Multi-stage schemes derived from high-order A-stable Pad´e methods are ideally suited to deal with the temporal aspect of such problems. However, when such temporal schemes are combined with the standard Galerkin finite element method for spatial discretization, the resulting schemes typically suffer from a lack of numerical dissipation in situations of purely convective transport. To obtain more stable approximations the present paper introduces a least-squares formulation that can be applied as an alternative to the traditional Galerkin formulation with the advantage of producing a symmetric positive algebraic system. In this manner numerical dissipation can be added to non-dissipative time-stepping algorithms and the approach represents a generalization to high-order accurate Pad´ e temporal schemes of the least-squares formulations previously developed by Carey and Jiang and Park and Liggett in connection with the Crank-Nicolson time-stepping scheme. The paper includes a study of the accuracy properties of the developed Pad´e/Leastsquares finite element schemes in comparison with corresponding Pad´e/Galerkin schemes. The modified equation method of Warming and Hyett is used to provide information about the dissipation and dispersion properties of the numerical schemes. The paper closes with the presentation of some illustrative examples highlighting the effectiveness of the least-squares approach in stabilizing finite element solutions to unsteady convective transport problems involving strong gradients.
