Ultraconvergence of ZZ Patch Recovery at Mesh Symmetry Points

Ultraconvergence property of the Zienkiewicz-Zhu gradient patch recovery technique based on local discrete least squares fitting is established for a large class of even-order finite elements. The result is valid at all rectangular mesh symmetry points. Different smoothing strategies are discussed. Superconvergence recovery for the Q8 element is proved and ultraconvergence numerical examples are demonstrated

A Meshless Gradient Recovery Method Part I: Superconvergence Property

A new gradient recovery method is introduced and analyzed. It is proved that the method is superconvergent for translation invariant finite element spaces of any order. The method maintains the simplicity, efficiency, and superconvergence properties of the Zienkiewicz-Zhu patch recovery method. In addition, under uniform triangular meshes, the method is superconvergent for the Chevron pattern, and ultraconvergence at element edge centers for the regular pattern

Recovery Techniques For Finite Element Methods And Their Applications

Recovery techniques are important post-processing methods to obtain improved approximate solutions from primary data with reasonable cost. The practical us- age of recovery techniques is not only to improve the quality of approximation, but also to provide an asymptotically exact posteriori error estimators for adaptive meth- ods. This dissertation presents recovery techniques for nonconforming finite element methods and high order derivative as well as applications of gradient recovery. Our first target is to develop a systematic gradient recovery technique for Crouzeix- Raviart element. The proposed method uses finite element solution to build a better approximation of the exact gradient based on local least square fittings. Due to poly- nomial preserving property of least square fitting, it is easy to show that the new proposed method preserves quadratic polynomials. In addition, the proposed gra- dient recovery is linearly bounded. Numerical tests indicate the recovered gradient is superconvergent to the exact gradient for both second order elliptic equation and Stokes equation. The gradient recovery technique can be used in a posteriori error estimates for Crouzeix-Raviart element, which is relatively simple to implement and problem independent. Our second target is to propose and analyze a new effective Hessian recovery for continuous finite element of arbitrary order. The proposed Hessian recovery is based on polynomial preserving recovery. The proposed method preserves polynomials of degree (k + 1) on general unstructured meshes and polynomials of degree (k + 2) on translation invariant meshes. Based on it polynomial preserving property, we can able to prove superconvergence of the proposed method on mildly structured meshes. In addition, we establish the ultraconvergence result for the new Hessian recovery technique on translation invariant finite element space of arbitrary order. Our third target is to demonstrate application of gradient recovery in eigenvalue computation. We propose two superconvergent two-grid methods for elliptic eigen- value problems by taking advantage of two-gird method, two-space method, shifted- inverse power method, and gradient recovery enhancement. Theoretical and numer- ical results reveal that the proposed methods provide superconvergent eigenfunction approximation and ultraconvergent eigenvalue approximation. In addition, two mul- tilevel adaptive methods based recovery type a posterior error estimate are proposed

h-Adaptive finite element method: extension of the isotropic error density recovery remeshing strategy of quadratic order

Orientador: Prof. Dr. Jucélio Tomás PereiraDissertação (mestrado) - Universidade Federal do Paraná, Setor de Tecnologia, Programa de Pós-Graduação em Engenharia Mecânica. Defesa : Curitiba, 05/07/2018Inclui referências: p.86-90Área de concentração: Mecânica dos Sólidos e VibraçõesResumo: O Método de Elementos Finitos (MEF) é uma técnica para resolver numericamente problemas físicos comumente utilizada na engenheria. Um fator importante na obtenção de uma solução precisa e eficiente decorre da utilização adequada da malha de discretização. Tipicamente, técnicas h-adaptativas são empregadas para projeção de uma malha ótima, onde o erro estimado em cada elemento é distribuído e minimizado de acordo com um critério de malha ótima. Neste contexto, o presente trabalho estende e avalia o método de refino hadaptativo denominado de Recuperação da Densidade do Erro Isotrópica (IEDR) para elementos triangulares quadráticos. Inicialmente desenvolvida para elementos lineares, esta técnica baseia-se na recuperação de uma função densidade do erro em energia em conjunto com a solução de um problema de otimização que busca o tamanho do novo elemento. Dessa maneira, a metodologia IEDR aborda os erros provenientes do MEF de maneira que contenha informações locais com maior abrangência, já que, nesta metodologia, uma função densidade do erro é recuperada. Os parâmetros de qualidade de malha, obtidos através desta técnica, são comparados à tradicionais técnicas de projeto de malha denominada de Chp e à técnica Li- Bettess (LB). A estimativa dos erros de discretização é realizada através do estimador de erro a posteriori baseado em recuperação, onde os gradientes recuperados são obtidos pelo método Superconvergente de Recuperação de Padrões (Superconvergent Patch Recovery - SPR). A implementação computacional é elaborada no software Matlab®, sendo a geração de malha realizada pelo gerador Bidimensional Anisotropic Mesh Generator (BAMG). Resultados numéricos demonstram que o processo h-adaptativo baseado na técnica IEDR obtém malhas convergentes para problemas com e sem singularidade, as quais apresentam, em geral, vantagens em relação ao número de graus de liberdade, à convergência e aos parâmetros de malha em comparação à tradicional técnica Chp e vantagens comparada à técnica LB para elementos quadráticos. Palavras-chave: Elemento Triangular de Deformações Lineares. h-adaptividade. Método dos Elementos Finitos. Estimadores de erro a posteriori. Recuperação da Densidade do Erro Isotrópica.Abstract: The finite element method (FEM) is a technique used to numerically solve physics problems which is often used in engineering. One factor in obtaining a solution that has acceptable accuracy is using adequate mesh discretization. Typically, h-adaptive techniques are used to determine new element sizes based on errors distributed among each element following an optimum mesh criterion. In this context, the current work proposes to extend and analyze the Isotropic Error Density Recovery (IEDR) h-refinement method for quadratic triangular finite elements. Initially developed for linear triangular finite elements, the extended technique is based on the recovery of an error density function, such that an optimization technique is used to search for the new element sizes. Hence, the IEDR technique utilizes more information of the local errors to design element sizes due to the recovery of an element error density function. The h-adaptive finite element method process based on the IEDR technique is compared to the traditionally used Chp and Li-Bettess mesh design techniques found in the literature. The discretization error estimates are achieved via a recovery based a posteriori error estimator, whereas the recovered gradients are obtained using the Superconvergent Patch Recovery Method. The algorithm is implemented using Matlab®, while the mesh generation is done by the Bidimensional Anisotropic Mesh Generator (BAMG). Results show, overall, that the meshes designed through the proposed methodology obtain superior mesh quality parameters, less degrees of freedom and better convergence in comparison with the traditional Chp remeshing methodology and advantages compared to the Li-Bettess element size estimation technique for quadratic elements. Keywords: Linear Strain Triangle. h-adaptativity. Finite Element Method. a posteriori Error Estimates. Isotropic Error Density Recovery

Orthogonality constrained gradient reconstruction for superconvergent linear functionals

The post-processing of the solution of variational problems discretized with Galerkin finite element methods is particularly useful for the computation of quantities of interest. Such quantities are generally expressed as linear functionals of the solution and the error of their approximation is bounded by the error of the solution itself. Several a posteriori recovery procedures have been developed over the years to improve the accuracy of post-processed results. Nonetheless such recovery methods usually deteriorate the convergence properties of linear functionals of the solution and, as a consequence, of the quantities of interest as well. The paper develops an enhanced gradient recovery scheme able to both preserve the good qualities of the recovered gradient and increase the accuracy and the convergence rates of linear functionals of the solution

Analysis of Recovery Type A Posteriori Error Estimators for Mildly Structured Grids

Some recovery type error estimators for linear finite element method are analyzed under O(h1+alpha) (alpha greater than 0) regular grids. Superconvergence is established for recovered gradients by three different methods when solving general non-self-adjoint second-order elliptic equations. As a consequence, a posteriori error estimators based on those recovery methods are asymptotically exact