Improved ZZ A Posteriori Error Estimators for Diffusion Problems: Conforming Linear Elements

In \cite{CaZh:09}, we introduced and analyzed an improved Zienkiewicz-Zhu (ZZ) estimator for the conforming linear finite element approximation to elliptic interface problems. The estimator is based on the piecewise "constant" flux recovery in the $H(div;\Omega)$ conforming finite element space. This paper extends the results of \cite{CaZh:09} to diffusion problems with full diffusion tensor and to the flux recovery both in piecewise constant and piecewise linear $H(div)$ space.Comment: arXiv admin note: substantial text overlap with arXiv:1407.437

ZZ-type aposteriori error estimators for adaptive boundary element methods on a curve

In the context of the adaptive finite element method (FEM), ZZ-error estimators named after Zienkiewicz and Zhu are mathematically well-established and widely used in practice. In this work, we propose and analyze ZZ-type error estimators for the adaptive boundary element method (BEM). We consider weakly-singular and hyper-singular integral equations and prove, in particular, convergence of the related adaptive mesh-refining algorithms

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

Reliable averaging for the primal variable in the Courant FEM and hierarchical error estimators on red-refined meshes

A hierarchical a posteriori error estimator for the first-order finite element method (FEM) on a red-refined triangular mesh is presented for the 2D Poisson model problem. Reliability and efficiency with some explicit constant is proved for triangulations with inner angles smaller than or equal to pi/2. The error estimator does not rely on any saturation assumption and is valid even in the pre-asymptotic regime on arbitrarily coarse meshes. The evaluation of the estimator is a simple post-processing of the piecewise linear FEM without any extra solve plus a higher-order approximation term. The results also allows the striking observation that arbitrary local averaging of the primal variable leads to a reliable and efficient error estimation. Several numerical experiments illustrate the performance of the proposed a posteriori error estimator for computational benchmarks

Reliable averaging for the primal variable in the Courant FEM and hierarchical error estimators on red-refined meshes

A hierarchical a posteriori error estimator for the first-order finite element method (FEM) on a red-refined triangular mesh is presented for the 2D Poisson model problem. Reliability and efficiency with some explicit constant is proved for triangulations with inner angles smaller than or equal to π/2 . The error estimator does not rely on any saturation assumption and is valid even in the pre-asymptotic regime on arbitrarily coarse meshes. The evaluation of the estimator is a simple post-processing of the piecewise linear FEM without any extra solve plus a higher-order approximation term. The results also allows the striking observation that arbitrary local averaging of the primal variable leads to a reliable and efficient error estimation. Several numerical experiments illustrate the performance of the proposed a posteriori error estimator for computational benchmarks

Real-time Error Control for Surgical Simulation

Objective: To present the first real-time a posteriori error-driven adaptive finite element approach for real-time simulation and to demonstrate the method on a needle insertion problem. Methods: We use corotational elasticity and a frictional needle/tissue interaction model. The problem is solved using finite elements within SOFA. The refinement strategy relies upon a hexahedron-based finite element method, combined with a posteriori error estimation driven local $h$-refinement, for simulating soft tissue deformation. Results: We control the local and global error level in the mechanical fields (e.g. displacement or stresses) during the simulation. We show the convergence of the algorithm on academic examples, and demonstrate its practical usability on a percutaneous procedure involving needle insertion in a liver. For the latter case, we compare the force displacement curves obtained from the proposed adaptive algorithm with that obtained from a uniform refinement approach. Conclusions: Error control guarantees that a tolerable error level is not exceeded during the simulations. Local mesh refinement accelerates simulations. Significance: Our work provides a first step to discriminate between discretization error and modeling error by providing a robust quantification of discretization error during simulations.Comment: 12 pages, 16 figures, change of the title, submitted to IEEE TBM

Self-Adaptive Methods for PDE

[no abstract available

Gradient recovery in adaptive finite element methods for parabolic problems

We derive energy-norm aposteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error, for fully discrete schemes for the linear heat equation. This appears to be the first completely rigorous derivation of ZZ estimators for fully discrete schemes for evolution problems, without any restrictive assumption on the timestep size. An essential tool for the analysis is the elliptic reconstruction technique. Our theoretical results are backed with extensive numerical experimentation aimed at (a) testing the practical sharpness and asymptotic behaviour of the error estimator against the error, and (b) deriving an adaptive method based on our estimators. An extra novelty provided is an implementation of a coarsening error "preindicator", with a complete implementation guide in ALBERTA.Comment: 6 figures, 1 sketch, appendix with pseudocod