A posteriori error estimates for an hp finite element method

MARÍA G ARMENTANO, CLAUDIO PADRA, RODOLFO RODRIGUEZ, MARIO SCHEBLE

Valparaiso

Congreso; Valparaiso Numerico IV; 2013

In this talk we first introduce an hp finite element method to solve two-dimensionalfluid-structure spectral problems in polygonal domains, which arise from the computation of the vibration modes of a bundle of parallel tubes immersed in an incompressible fluid cite{APRS,APRS2,COP}. We prove the convergence of the method and we define an a posteriori error estimator of the residual type which can be computed locally from theapproximate eigenpair. We show its reliability and efficiency by proving that the  estimator is equivalent to the energy norm of the error up to higher order terms, the equivalence constant of the efficiency estimate being suboptimal in that it depends on the  polynomial degree. Following the hp adaptive strategy given in cite{MW} we present  an hp adaptive algorithm and several numerical tests which show the performance of  the scheme, including some numerical evidence of exponential convergence. On the other hand, we also present an hp finite element adaptive scheme to solve the  Poisson problem on curved domains. We show the advantage of using curved triangles  cite{Z1} and we exhibit the loss of convergence rate if we use standard triangular elements with straight edges cite{APS}.