A Posteriori Error Estimation for an Interior Penalty Type Method Employing $H(\mathrm{div})$ Elements for the Stokes Equations

In many scientific problems, adaptive finite element methods has been widely used to improve the accuracy of numerical solutions. The general idea is to refine or adjust the mesh such that the errors are “equally” distributed over the computational mesh, with the aim of achieving a better accurate solution using an optimal number of degrees of freedom. By using the information from the approximated solution and the known data, the a posteriori error estimator provides the information about the size and the distribution of the error of the finite element approximation. There is a large numerical analysis literature on adaptive finite element methods, and various types of a posteriori estimates have been proposed for different problems, see e.g. [1]. The a posterior error estimate and adaptive finite element method were first introduced by [2]. Since the later 1980s, much research work on a posteriori error estimate has been developed including the residual type a posteriori error estimate [8], recovery type a posteriori error estimate [16], a posteriori error estimate based on hierarchic basis [5, 4], and so on. For the literature, the readers are referred to the books [1, 3, 12, 14], the papers [6, 13, 15], and the references cited therein.KeywordsError EstimatorPosteriori ErrorPosteriori Error EstimateInitial MeshElement Stiffness MatrixThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The thesis deals with theoretical and applicative aspects of some innovative numerical techniques for the simulation of the flow in Discrete Fracture Networks (DFN). In particular, the recently developed Virtual Element Method (VEM) is considered. A VEM-SUPG stabilized formulation for advection-diffusion problems is defined and studied theoretically and numerically, as well as a residual a posteriori error estimate which does not include any term depending on the VEM stabilization form. Regarding DFN flow simulations, an approach based on Virtual Elements and standard domain decomposition techniques such as Mortar methods is introduced and studied, also in combination with the use of orthogonal polynomials to avoid numerical instabilities that arise when computing polynomial projections on very badly shaped elements. Finally, we consider a constrained optimization formulation of the problem of computing the flow in DFNs and we develop a residual based a posteriori error estimate that contains non standard terms related to the geometrical non-conformity of the mesh on each fracture to the intersections between them.

A simplified stochastic Hookean dumbbells model arising from viscoelastic flows is considered, the convective terms being disregarded. A finite element discretization in space is proposed. Existence of the numerical solution is proved for small data, so as a priori error estimates, using an implicit function theorem and regularity results obtained in [Bonito et al., J. Evol. Equ. 6 (2006) 381–398] for the solution of the continuous problem. A posteriori error estimates are also derived. Numerical results with small time steps and a large number of realizations confirm the convergence rate with respect to the mesh size.

A posteriori error estimates of two-grid finite volume element methods for nonlinear elliptic problems

We derive residual‐based a posteriori error estimates of finite element method for linear parabolic interface problems in a two‐dimensional convex polygonal domain. Both spatially discrete and fully discrete approximations are analyzed. While the space discretization uses finite element spaces that are allowed to change in time, the time discretization is based on the backward Euler approximation. The main ingredients used in deriving a posteriori estimates are new Clément type interpolation estimates and an appropriate adaptation of the elliptic reconstruction technique introduced by (Makridakis and Nochetto, SIAM J Numer Anal 4 (2003), 1585–1594). We use only an energy argument to establish a posteriori error estimates with optimal order convergence in the ‐norm and almost optimal order in the ‐norm. The interfaces are assumed to be of arbitrary shape but are smooth for our purpose. Numerical results are presented to validate our derived estimators. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 570–598, 2017

On a posteriori error estimation using distances between numerical solutions and angles between truncation errors

The practice of engineering calculation shows that the adaptive finite element method is quite an efficient numerical method for solving partial differential equations, where the reliable and effective a posteriori error estimator plays a crucial role in the implementation.In this talk, we first design a weak gradient recovery type a posteriori error estimator of the weak Galerkin finite element method for the elliptic equation.Then, for the Crank-Nicolson weak Galerkin finite element numerical discretization of the linear parabolic equation, we establish its time-space a posteriori error estimation by the elliptic reconstruction technique, where the numerical error is split into the elliptic error and the parabolic error by introducing the corresponding elliptic equation.The theory shows that any a posteriori error estimator of the elliptic equation can be taken to control the elliptic error, and the error of numerical solutions between two adjacent time layers can be used to control the parabolic error.The rationality of the error estimators is verified numerically according to the time-space adaptive algorithm.Finally, the conclusion of the linear parabolic equation is extended to the SAV weak Galerkin finite element discretization of the Allen-Cahn equation.Exploiting the time-space adaptive algorithm, the dynamic interface evolution of the Allen-Cahn equation is simulated by some benchmark numerical examples, which validates the effectiveness of the time-space error estimators.

For elliptic interface problems in two and three dimensions, this paper studies a priori and residual-based a posteriori error estimations for the Crouzeix--Raviart nonconforming and the discontinuous Galerkin finite element approximations. It is shown that both the a priori and the a posteriori error estimates are robust with respect to the diffusion coefficient, i.e., constants in the error bounds are independent of the jump of the diffusion coefficient. The a priori estimates are also optimal with respect to local regularity of the solution. Moreover, we obtained these estimates with no assumption on the distribution of the diffusion coefficient.

The workshop Nonstandard Finite Element Methods , organized by Susanne C. Brenner (Baton Rouge), Carsten Carstensen (Berlin) and Peter Monk (Newark) was held August 10 – 16, 2008. This meeting was well attended with over 40 participants with broad geographic representation. Although Courant is often credited with the discovery of finite elements [2, 3], the first practical use of such methods in engineering seems to date back to the work of Argyris and Clough et. al. (apparently Clough coined the name “finite elements”) in the 1950s [1, 4]. By the 1970s the finite element method was firmly established in engineering practice and the basic theory of conforming elements for elliptic problems was well understood [2]. Engineers and mathematicians have since expanded the use of finite elements to a wide variety of new applications and have improved the theoretical underpinnings of the method. In addition, efficient computational algorithms have been developed, such as the multigrid method, that allow for the application of finite elements to sophisticated three dimensional problems. Within current research, we can distinguish classical or standard finite element methods, and newer nonstandard finite elements The former include conforming elements applied to “standard” Galerkin discretizations of problems. Usually accuracy is obtained by mesh refinement. As the range of applications has increased, and to alleviate perceived shortcomings in standard conforming elements, numerical analysts and engineers have sought more general finite element methods based, for example, on mixed formulations, and non-conforming or even discontinuous Galerkin schemes. These methods may be superior to standard finite elements in particular applications because of enhancements to stability, robustness or conservation properties. We refer to them as non-standard finite element methods. This Oberwolfach workshop was devoted to non-standard finite element methods and their analyses. Inevitability such a snapshot of the subject is biased, but we have tried to include non-standard methods in the broadest sense including mixed, discontinuous Galerkin, generalized, partition of unity and mortar FEM methods. All these schemes have in common that stability and convergence is not obvious and requires mathematical analysis. This is even more true for developing fast solvers, a posteriori error estimation and adaptive mesh design. Through the week there were 29 presentations on various non-standard topics. These provided an overview of current research directions, new developments and open problems.

An improved a posteriori error estimate for the Galerkin spectral method in one dimension

New a posteriori error estimator and adaptive mesh refinement strategy for 2-D crack problems

The paper deals with the application of an adaptive, hierarchic‐iterative finite element technique to solve two‐dimensional electromechanical boundary value problems with impermeable cracks in piezoelectric plates. In order to compute the dielectric and mechanical intensity factors, the interaction integral technique is used. The iterative finite element solver takes advantage of a sequence of solutions on hierarchic discretizations. Based on an a posteriori error estimation, the finite element mesh is locally refined or coarsened in each step. Two crack configurations are investigated in an infinite piezoelectric plate: A finite straight crack and a finite kinked crack. Fast convergence of the numerical intensity factors to the corresponding analytical solution is exemplarily proved during successive adaptive steps for the first configuration. Similar tendency can be observed for the second configuration. Furthermore, the computed intensity factors for the kinks are found to coincide well with the corresponding analytical values. In order to simulate the kinks spreading from a straight crack, the finite element mesh is modified automatically with a specially developed algorithm. This forms the basis for a fully adaptive simulation of crack propagation. Copyright © 2009 John Wiley & Sons, Ltd.

We design and analyze a C^0 interior penalty method for the approximation of classical solutions of the Dirichlet boundary value problem of the Monge–Ampère equation on convex polygonal domains. The method is based on an enhanced cubic Lagrange finite element that enables the enforcement of the convexity of the approximate solutions. Numerical results that corroborate the a priori and a posteriori error estimates are presented. It is also observed from numerical experiments that this method can capture certain weak solutions.

Superconvergence property of an over-penalized discontinuous Galerkin finite element gradient recovery method

Residual-based a posteriori error estimates were derived within one unifying framework for lowest-order conforming, nonconforming, and mixed finite element schemes in Carstensen [Numer Math 100:617–637, 2005]. Therein, the key assumption is that the conforming first-order finite element space $$V_h^c$$ annulates the linear and bounded residual l written $$V_h^c \subseteq {\rm ker} \ell$$. That excludes particular nonconforming finite element methods (NCFEMs) on parallelograms in that $$V_h^c \not\subset {\rm ker} \ell$$. The present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates, also for mortar and discontinuous Galerkin methods. The key assumption is the existence of some bounded linear operator $$\Pi: V_h^c \rightarrow V_h^{nc}$$ with some elementary properties. It is conjectured that the more general hypothesis (H1)–(H3) can be established for all known NCFEMs. Applications on various nonstandard finite element schemes for the Laplace, Stokes, and Navier–Lame equations illustrate the presented unifying theory of a posteriori error control for NCFEM.