An adaptive strategy for the bivariate solution of finite element problems using multivariate nested Padé approximants

Abstract

SUMMARYMost engineering applications involving solutions by numerical methods are dependent on several parameters, whose impact on the solution may significantly vary from one to the other. At times an evaluation of these multivariate solutions may be required at the expense of a prohibitively high computational cost. In the present paper, an adaptive approach is proposed as a way to estimate the solution of such multivariate finite element problems. It is based upon the integration of so‐called nested Padé approximants within the finite element procedure. This procedure includes an effective control of the approximation error, which enables adaptive refinements of the converged intervals upon reconstruction of the solution. The main advantages lie in a potential reduction of the computational effort and the fact that the level of a priori knowledge required about the solution in order to have an accurate, efficient, and well‐sampled estimate of the solution is low. The approach is introduced for bivariate problems, for which it is validated on elasto‐poro‐acoustic problems of both academic and more industrial scale. It is argued that the methodology in general holds for more than two variables, and a discussion is opened about the truncation refinements required in order to generalize the results accordingly. Copyright © 2014 John Wiley & Sons, Ltd.