Everything you always wanted to know about a-posteriori error estimation in finite element methods, but were afraid to ask

In this paper the basic concepts to obtain a posteriori error estimates for the finite element method are reviewed. Explicit residual-based, implicit (namely subdomain and element) residual-based, hierarchical-based, recovery-base and functional-based error estimators as well as goal oriented error estimators are presented for a test elliptic boundary value problem. These notes are an introductory presentation, reviewing in a not-too-technical way the fundamental concepts involved in the subject and do not aim at being exhaustive or complete but rather simple and easy to follow. For more detailed explanations, we refer the interested reader to [3] and eventually to [4],[9],[15],[27],[30], chapter 4 of [37],[38],[39] – and the references therein – where most of the material contained in this report can be found

A posteriori error estimation in the finite element method

The work broadly consists of two parts. In the first part we construct a framework for analyzing and developing a posteriori error estimators for use in the finite element solution of elliptic partial differential equations which have smooth solutions. The analysis makes use of complementary variational principles and the superconvergence phenomenon associated with the finite element method. The second part generalizes these results to the important case when the solution of the boundary value problem contains singularities. It is shown how the classical techniques may be easily modified to perform satisfactorily for the singular case

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

Polynomial Preserving Recovery For Weak Galerkin Methods And Their Applications

Gradient recovery technique is widely used to reconstruct a better numerical gradient from a finite element solution, for mesh smoothing, a posteriori error estimate and adaptive finite element methods. The PPR technique generates a higher order approximation of the gradient on a patch of mesh elements around each mesh vertex. It can be used for different finite element methods for different problems. This dissertation presents recovery techniques for the weak Galerkin methods and as well as applications of gradient recovery on various of problems, including elliptic problems, interface problems, and Stokes problems. Our first target is to develop a boundary strategy for the current PPR algorithm. The current accuracy of PPR near boundaries is not as good as that in the interior of the domain. It might be even worse than without recovery. Some special treatments are needed to improve the accuracy of PPR on the boundary. In this thesis, we present two boundary recovery strategies to resolve the problem caused by boundaries. Numerical experiments indicate that both of the newly proposed strategies made an improvement to the original PPR. Our second target is to generalize PPR to the weak Galerkin methods. Different from the standard finite element methods, the weak Galerkin methods use a different set of degrees of freedom. Instead of the weak gradient information, we are able to obtain the recovered gradient information for the numerical solution in the generalization of PPR. In the PPR process, we are also able to recover the function value at the nodal points which will produce a global continuous solution instead of piecewise continuous function. Our third target is to apply our proposed strategy and WGPPR to interface problems. We treat an interface as a boundary when performing gradient recovery, and the jump condition on the interface can be well captured by the function recovery process. In addition, adaptive methods based on WGPPR recovery type a posteriori error estimator is proposed and numerically tested in this thesis. Application on the elliptic problem and interface problem validate the effectiveness and robustness of our algorithm. Furthermore, WGPPR has been applied to 3D problem and Stokes problem as well. Superconvergent phenomenon is again observed

Recovery type a posteriori error estimation of an adaptive finite element method for Cahn--Hilliard equation

In this paper, we derive a novel recovery type a posteriori error estimation of the Crank-Nicolson finite element method for the Cahn--Hilliard equation. To achieve this, we employ both the elliptic reconstruction technique and a time reconstruction technique based on three time-level approximations, resulting in an optimal a posteriori error estimator. We propose a time-space adaptive algorithm that utilizes the derived a posteriori error estimator as error indicators. Numerical experiments are presented to validate the theoretical findings, including comparing with an adaptive finite element method based on a residual type a posteriori error estimator.Comment: 36 pages, 7 figure

A posteriori error estimator based on gradient recovery by averaging for convection-diffusion-reaction problems approximated by discontinuous Galerkin methods

International audienceWe consider some (anisotropic and piecewise constant) convection-diffusion-reaction problems in domains of R2, approximated by a discontinuous Galerkin method with polynomials of any degree. We propose two a posteriori error estimators based on gradient recovery by averaging. It is shown that these estimators give rise to an upper bound where the constant is explicitly known up to some additional terms that guarantees reliability. The lower bound is also established, one being robust when the convection term (or the reaction term) becomes dominant. Moreover, the estimator is asymptotically exact when the recovered gradient is superconvergent. The reliability and efficiency of the proposed estimators are confirmed by some numerical tests