An alternative approach to the Isotropic Error Density Recovery Method based on local patches

Frederico Alves Jahnert¹; Jucélio Tomás Pereira¹; Jéderson da Silva¹

¹ Universidade Federal do Paraná - UFPR

doi:10.20906/CPS/CILAMCE2017-1016

Resumo

In the Finite Element Method, regarding the control and limitation of approximation errors contained in the solution, an important factor to consider involves determining an adequate mesh discretization. In this context, the h-adaptive Finite Element method based on a posteriori error estimation has demonstrated to be a valuable tool. Hence, this study aims at obtaining an efficient mesh refinement methodology by proposing a modification to the Isotropic Error Density Recovery h-adaptive mesh refinement method (IEDR - Pereira et al., 2016). Through the use of a posteriori error estimator based on recovery, the aforementioned technique evaluates a quadratic error in energy density function and calculates the analytical solution of an optimization problem via the Lagrangian Method to find an expression for the size of the new element. Originally, the IEDR technique was proposed and analyzed for triangular finite elements with linear approximation. In addition, the originally proposed methodology evaluates the error in energy density function utilizing the centroid of each element as origin. Consequentially, the new element size is estimated on that origin location even though the mesh parameters are defined through nodal metrics, which requires post-processing of the estimated dimensions. For these reasons, the current article proposes a definition of the quadratic error in energy density function with origin at each node of the mesh. This is carried out by evaluating the function at internal points located in a patch of elements formed by all the elements connected to the node to be evaluated. This new methodology, named here as Isotropic Error Density Nodal Recovery (IEDNR) offers a way to evaluate the new element parameters directly in each node of the mesh. Also, based on the results obtained, the IEDNR methodology presents itself as an approach to generalize the IEDR technique for any polynomial order of approximation. The efficiency of the IEDNR methodology is analyzed in elliptic two-dimensional problems. Th

Palavras-chave: Finite Element Method; h-adaptativity; Superconvergent Patch Recovery; Isotropic Error Density Recovery