A posteriori error estimates of mixed DG finite element methods for linear parabolic equations

In this article, we analyse a posteriori error estimates of mixed finite element discretizations for linear parabolic equations. The space discretization is done using the order λ?≥?1 Raviart–Thomas mixed finite elements, whereas the time discretization is based on discontinuous Galerkin (DG) methods (r?≥?1). Using the duality argument, we derive a posteriori l ^∞(L ²) error estimates for the scalar function, assuming that only the underlying mesh is static.     

Uniform convergence and a posteriori error estimators for the enhanced strain finite element method

Summary. Enhanced strain elements, frequently employed in practice, are known to improve the approximation of standard (non-enhanced) displacement-based elements in finite element computations. The first contribution in this work towards a complete theoretical explanation for this observation is a proof of robust convergence of enhanced element schemes: it is shown that such schemes are locking-free in the incompressible limit, in the sense that the error bound in the a priori estimate is independent of the relevant Lamé constant. The second contribution is a residual-based a posteriori error estimate; the L ² norm of the stress error is estimated by a reliable and efficient estimator that can be computed from the residuals. Mathematics Subject Classification (2000):65N30     

A posteriori error estimates for the Crank-Nicolson method for parabolic equations

We derive optimal order a posteriori error estimates for time discretizations by both the Crank-Nicolson and the Crank-Nicolson-Galerkin methods for linear and nonlinear parabolic equations. We examine both smooth and rough initial data. Our basic tool for deriving a posteriori estimates are second-order Crank-Nicolson reconstructions of the piecewise linear approximate solutions. These functions satisfy two fundamental properties: (i) they are explicitly computable and thus their difference to the numerical solution is controlled a posteriori, and (ii) they lead to optimal order residuals as well as to appropriate pointwise representations of the error equation of the same form as the underlying evolution equation. The resulting estimators are shown to be of optimal order by deriving upper and lower bounds for them depending only on the discretization parameters and the data of our problem. As a consequence we provide alternative proofs for known a priori rates of convergence for the Crank-Nicolson method.

     

A posteriori error analysis for time-dependent Ginzburg-Landau type equations

Summary. This work presents an a posteriori error analysis for the finite element approximation of time-dependent Ginzburg-Landau type equations in two and three space dimensions. The solution of an elliptic, self-adjoint eigenvalue problem as a post-processing procedure in each time step of a finite element simulation leads to a fully computable upper bound for the error. Theoretical results for the stability of degree one vortices in Ginzburg-Landau equations and of generic interfaces in Allen-Cahn equations indicate that the error estimate only depends on the inverse of a small parameter in a low order polynomial. The actual dependence of the error estimate upon this parameter is explicitly determined by the computed eigenvalues and can therefore be monitored within an approximation scheme. The error bound allows for the introduction of local refinement indicators which may be used for adaptive mesh and time step size refinement and coarsening. Numerical experiments underline the reliability of this approach.Mathematics Subject Classification(2000): 65M15, 65M60, 65M50.AcknowledgmentS.B. is thankful to G. Dolzmann and R.H. Nochetto for stimulating discussions. This work was supported by a fellowship within the Postdoc-Programme of the German Academic Exchange Service (DAAD).     

Implicit a posteriori error estimates for the Maxwell equations

An implicit a posteriori error estimation technique is presented and analyzed for the numerical solution of the time-harmonic Maxwell equations using Nédélec edge elements. For this purpose we define a weak formulation for the error on each element and provide an efficient and accurate numerical solution technique to solve the error equations locally. We investigate the well-posedness of the error equations and also consider the related eigenvalue problem for cubic elements. Numerical results for both smooth and non-smooth problems, including a problem with reentrant corners, show that an accurate prediction is obtained for the local error, and in particular the error distribution, which provides essential information to control an adaptation process. The error estimation technique is also compared with existing methods and provides significantly sharper estimates for a number of reported test cases.

     

A posteriori error estimators for a two-level finite element method for the Navier-Stokes equations

Two- and multilevel truncated Newton finite element discretizations are presently a very promising approach for approximating the (nonlinear) Navier-Stokes equations describing the equilibrium flow of a viscous, incompressible fluid. Their combination with mesh adaptivity is considered in this article. Specifically, locally calculable a posteriori error estimators are derived, with full mathematical support, for the basic two-level discretization of the Navier-Stokes equations. © 1996 John Wiley & Sons, Inc.     

A posteriori error analysis for a class of integral equations and variational inequalities

We consider elliptic and parabolic variational equations and inequalities governed by integro-differential operators of order ${2s \in (0,2]}We consider elliptic and parabolic variational equations and inequalities governed by integro-differential operators of order 2s ? (0,2]{2s \in (0,2]}. Our main motivation is the pricing of European or American options under Lévy processes, in particular pure jump processes or jump diffusion processes with tempered stable processes. The problem is discretized using piecewise linear finite elements in space and the implicit Euler method in time. We construct a residual-type a posteriori error estimator which gives a computable upper bound for the actual error in H ^s -norm. The estimator is localized in the sense that the residuals are restricted to the discrete non-contact region. Numerical experiments illustrate the accuracy of the space and time estimators, and show that they can be used to measure local errors and drive adaptive algorithms.     

Optimal error estimate and superconvergence of the DG method for first-order hyperbolic problems

We consider the original discontinuous Galerkin method for the first-order hyperbolic problems in d-dimensional space. We show that, when the method uses polynomials of degree k, the L₂-error estimate is of order k+1 provided the triangulation is made of rectangular elements satisfying certain conditions. Further, we show the O(h^2k+1)-order superconvergence for the error on average on some suitably chosen subdomains (including the whole domain) and their outflow faces. Moreover, we also establish a derivative recovery formula for the approximation of the convection directional derivative which is superconvergent with order k+1.     

Pointwise a posteriori error estimates for monotone semi-linear equations

We derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi-linear equations. The estimates hold for Lagrange elements of any fixed order, non-smooth nonlinearities, and take numerical integration into account. The proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately, and then applying the continuous maximum principle; no geometric mesh constraints are thus required. Numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature.     

A posteriori error estimators for the Stokes equations

Summary We present two a posteriori error estimators for the mini-element discretization of the Stokes equations. One is based on a suitable evaluation of the residual of the finite element solution. The other one is based on the solution of suitable local Stokes problems involving the residual of the finite element solution. Both estimators are globally upper and locally lower bounds for the error of the finite element discretization. Numerical examples show their efficiency both in estimating the error and in controlling an automatic, self-adaptive mesh-refinement process. The methods presented here can easily be generalized to the Navier-Stokes equations and to other discretization schemes.This work was accomplished at the Universität Heidelberg with the support of the Deutsche Forschungsgemeinschaft     

A posteriori error estimators for convection-diffusion equations

Summary. We derive a posteriori error estimators for convection-diffusion equations with dominant convection. The estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local mesh-Peclet number. The estimators are either based on the evaluation of local residuals or on the solution of discrete local Dirichlet or Neumann problems. Received February 10, 1997 / Revised version received November 4, 1997     

A posteriori error indicators for Maxwell''s equations

In this paper we shall analyze a class of a posteriori error indicators for an electromagnetic scattering problem for Maxwell's equations in the presence of a bounded, inhomogeneous and anisotropic scatterer. Problems of this type arise when computing the interaction of electromagnetic radiation with biological tissue. We briefly recall existence and uniqueness theory associated with this problem. Then we show how a posteriori error indicators can be derived using an adjoint equation approach. The error indicators use both the jump in normal and tangential components of the field across faces in the mesh.     

A posteriori error estimates for linear equations

Summary This paper describes upper and lowerp-norm error bounds for approximate solutions of the linear system of equationsAx=b. These bounds imply that the error is proportional to the quantity wherer is the residual andq is the conjugate index top. The constant of proportionality is larger than 1 and lies in a specified range. Similar results are obtained for approximations toA ^–1 and solutions of nonsingular linear equations on general spaces.Research was partially supported by NSF Grant DMS8901477     

A posteriori error estimates for Maxwell equations

Maxwell equations are posed as variational boundary value problems in the function space and are discretized by Nédélec finite elements. In Beck et al., 2000, a residual type a posteriori error estimator was proposed and analyzed under certain conditions onto the domain. In the present paper, we prove the reliability of that error estimator on Lipschitz domains. The key is to establish new error estimates for the commuting quasi-interpolation operators recently introduced in J. Schöberl, Commuting quasi-interpolation operators for mixed finite elements. Similar estimates are required for additive Schwarz preconditioning. To incorporate boundary conditions, we establish a new extension result.

     

A posteriori error estimates of stabilized finite volume method for the Stokes equations

In this work, the residual‐type posteriori error estimates of stabilized finite volume method are studied for the steady Stokes problem based on two local Gauss integrations. By using the residuals between the source term and numerical solutions, the computable global upper and local lower bounds for the errors of velocity in H¹ norm and pressure in L² norm are derived. Furthermore, a global upper bound of u ? u_h in L²‐norm is also derived. Finally, some numerical experiments are provided to verify the performances of the established error estimators. Copyright © 2015 John Wiley & Sons, Ltd.     

A posteriori error estimate for finite volume element method of the parabolic equations

In this article, residual‐type a posteriori error estimates are studied for finite volume element (FVE) method of parabolic equations. Residual‐type a posteriori error estimator is constructed and the reliable and efficient bounds for the error estimator are established. Residual‐type a posteriori error estimator can be used to assess the accuracy of the FVE solutions in practical applications. Some numerical examples are provided to confirm the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 259–275, 2017     

On the a posteriori error analysis for equations of prescribed mean curvature

We present two approaches to the a posteriori error analysis for prescribed mean curvature equations. The main difference between them concerns the estimation of the residual: without or with computable weights. In the second case, the weights are related to the eigenvalues of the underlying operator and thus provide local and computable information about the conditioning. We analyze the two approaches from a theoretical viewpoint. Moreover, we investigate and compare the performance of the derived indicators in an adaptive procedure. Our theoretical and practical results show that it is advantageous to estimate the residual in a weighted way.

     

Superconvergence analysis and a posteriori error estimation of a finite element method for an optimal control problem governed by integral equations

In this paper, we discuss the numerical simulation for a class of constrained optimal control problems governed by integral equations. The Galerkin method is used for the approximation of the problem. A priori error estimates and a superconvergence analysis for the approximation scheme are presented. Based on the results of the superconvergence analysis, a recovery type a posteriori error estimator is provided, which can be used for adaptive mesh refinement. The research project is supported by the National Basic Research Program under the Grant 2005CB321701 and the National Natural Science Foundation of China under the Grant 10771211.     

Improved maximum-norm a posteriori error estimates for linear and semilinear parabolic equations

Linear and semilinear second-order parabolic equations are considered. For these equations, we give a posteriori error estimates in the maximum norm that improve upon recent results in the literature. In particular it is shown that logarithmic dependence on the time step size can be eliminated. Semidiscrete and fully discrete versions of the backward Euler and of the Crank-Nicolson methods are considered. For their full discretizations, we use elliptic reconstructions that are, respectively, piecewise-constant and piecewise-linear in time. Certain bounds for the Green’s function of the parabolic operator are also employed.     

A posteriori error analysis of the heterogeneous multiscale method for homogenization problems

In this Note we derive a posteriori error estimates for a multiscale method, the so-called heterogeneous multiscale method, applied to elliptic homogenization problems. The multiscale method is based on a macro-to-micro formulation. The macroscopic method discretizes the physical problem in a macroscopic finite element space, while the microscopic method recovers the unknown macroscopic data on the fly during the macroscopic stiffness matrix assembly process. We propose a framework for the analysis allowing to take advantage of standard techniques for a posteriori error estimates at the macroscopic level and to derive residual-based indicators in the macroscopic domain for adaptive mesh refinement. To cite this article: A. Abdulle, A. Nonnenmacher, C. R. Acad. Sci. Paris, Ser. I 347 (2009).