A-posteriori Error Research Articles

A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift

In this contribution we address a-posteriori error estimation in $L^\infty(L^2)$ for a heterogeneous multiscale finite element approximation of time-dependent advection-diffusion problems with rapidly oscillating coefficient functions and with a large expected drift. Based on the error estimate, we derive an algorithm for an adaptive mesh refinement. The estimate and the algorithm are validated in numerical experiments, showing applicability and good results even for heterogeneous microstructures.

A-posteriori error estimation in maximum norm for a strongly coupled system of two singularly perturbed convection–diffusion problems

A system of two singularly perturbed convection–diffusion problems with strong coupling is studied. This system is discretized by an upwind finite difference scheme on an arbitrary non-uniform mesh. An a posteriori error estimation in the maximum norm is derived. Based on the a posteriori error estimation, a monitor function is proposed to design an adaptive grid algorithm. Especially, to implement the adaptive grid approach, we not only give a mesh generation algorithm, but also give a sub-algorithm, which is called an arc-length equidistribution algorithm. Numerical results are given to confirm our theoretical result.

Asymptotically optimal definite quadrature formulae of 4th order

We construct several sequences of asymptotically optimal definite quadrature formulae of fourth order and evaluate their error constants. Besides the asymptotical optimality, an advantage of our quadrature formulae is the explicit form of their weights and nodes. For the remainders of our quadrature formulae monotonicity properties are established when the integrand has a non-negative or non-positive fourth derivative, and a-posteriori error estimates are proven.

A-posteriori error analysis for lithium-ion concentrations in batteries utilizing the reduced-basis method

ABSTRACTIn this paper, the authors consider a parametrized non-linear parabolic differential equation, which is motivated by lithium-ion battery models. A standard finite volume discretization leads to a high-dimensional discrete non-linear problem so that simulation of the parametrized problem for various different parameters is very costly. Therefore, the reduced-basis method is applied, so that the number of degrees of freedom is reduced significantly and a fast numerical simulation of the model is possible. To control the error, an a-posteriori error estimator is derived. Numerical experiments show the efficiency of the approach.

Mass-conservative reconstruction of Galerkin velocity fields for transport simulations

Accurate calculation of mass-conservative velocity fields from numerical solutions of Richards’ equation is central to reliable surface–subsurface flow and transport modeling, for example in long-term tracer simulations to determine catchment residence time distributions. In this study we assess the performance of a local Larson-Niklasson (LN) post-processing procedure for reconstructing mass-conservative velocities from a linear (P1) Galerkin finite element solution of Richards’ equation. This approach, originally proposed for a-posteriori error estimation, modifies the standard finite element velocities by imposing local conservation on element patches. The resulting reconstructed flow field is characterized by continuous fluxes on element edges that can be efficiently used to drive a second order finite volume advective transport model. Through a series of tests of increasing complexity that compare results from the LN scheme to those using velocity fields derived directly from the P1 Galerkin solution, we show that a locally mass-conservative velocity field is necessary to obtain accurate transport results. We also show that the accuracy of the LN reconstruction procedure is comparable to that of the inherently conservative mixed finite element approach, taken as a reference solution, but that the LN scheme has much lower computational costs. The numerical tests examine steady and unsteady, saturated and variably saturated, and homogeneous and heterogeneous cases along with initial and boundary conditions that include dry soil infiltration, alternating solute and water injection, and seepage face outflow. Typical problems that arise with velocities derived from P1 Galerkin solutions include outgoing solute flux from no-flow boundaries, solute entrapment in zones of low hydraulic conductivity, and occurrences of anomalous sources and sinks. In addition to inducing significant mass balance errors, such manifestations often lead to oscillations in concentration values that can moreover cause the numerical solution to explode. These problems do not occur when using LN post-processed velocities.

A residual based snapshot location strategy for POD in distributed optimal control of linear parabolic equations

In this paper we study the approximation of a distributed optimal control problem for linear parabolic PDEs with model order reduction based on Proper Orthogonal Decomposition (POD-MOR). POD-MOR is a Galerkin approach where the basis functions are obtained upon information contained in time snapshots of the parabolic PDE related to given input data. In the present work we show that for POD-MOR in optimal control of parabolic equations it is important to have knowledge about the controlled system at the right time instances. For the determination of the time instances (snapshot locations) we propose an a-posteriori error control concept which is based on a reformulation of the optimality system of the underlying optimal control problem as a second order in time and fourth order in space elliptic system. Finally, we present numerical tests to illustrate our approach and to show the effectiveness of the method in comparison to existing approaches.

An a-posteriori error estimator for linear elastic fracture mechanics using the stable generalized/extended finite element method

In this study, a recovery-based a-posteriori error estimator originally proposed for the Corrected XFEM is investigated in the framework of the stable generalized FEM (SGFEM). Both Heaviside and branch functions are adopted to enrich the approximations in the SGFEM. Some necessary adjustments to adapt the expressions defining the enhanced stresses in the original error estimator are discussed in the SGFEM framework. Relevant aspects such as effectivity indexes, error distribution, convergence rates and accuracy of the recovered stresses are used in order to highlight the main findings and the effectiveness of the error estimator. Two benchmark problems of the 2-D fracture mechanics are selected to assess the robustness of the error estimator hereby investigated. The main findings of this investigation are: the SGFEM shows higher accuracy than G/XFEM and a reduced sensitivity to blending element issues. The error estimator can accurately capture these features of both methods.

Well-Balanced High Order 1D Schemes on Non-uniform Grids and Entropy Residuals

This paper is concerned with the construction of high order schemes on irregular grids for balance laws, including a discussion of an a-posteriori error indicator based on the numerical entropy production. We also impose well-balancing on non uniform grids for the shallow water equations, which can be extended similarly to other cases, obtaining schemes up to fourth order of accuracy with very weak assumptions on the regularity of the grid. Our results show the expected convergence rates, the correct propagation of shocks across grid discontinuities and demonstrate the improved resolution achieved with a locally refined non-uniform grid. The error indicator based on the numerical entropy production, previously introduced for the case of systems of conservation laws, is extended to balance laws. Its decay rate and its ability to identify discontinuities is illustrated on several tests. The schemes proposed in this work naturally can also be applied to systems of conservation laws.

An Optimal Adaptive Finite Element Method for an Obstacle Problem

Abstract This paper provides a refined a posteriori error control for the obstacle problem with an affine obstacle which allows for a proof of optimal complexity of an adaptive algorithm. This is the first adaptive mesh-refining finite element method known to be of optimal complexity for some variational inequality. The result holds for first-order conforming finite element methods in any spacial dimension based on shape-regular triangulation into simplices for an affine obstacle. The key contribution is the discrete reliability of the a posteriori error estimator from [Numer. Math. 107 (2007), 455–471] in an edge-oriented modification which circumvents the difficulties caused by the non-existence of a positive second-order approximation [Math. Comp. 71 (2002), 1405–1419].

An interior point method designed for solving linear quadratic optimal control problems with hp finite elements

We consider linear quadratic optimal control problems with elliptic partial differential equations. The problem is solved with an interior point method in the control variable. We prove convergence of this method in function space by employing a suitable smoothing operator. As discretization we choose hp-finite element method based on local estimates on the smoothness of functions. A fully adaptive algorithm is implemented and a-posteriori error estimators are derived for the central path and the Newton system. The theoretical results are complemented by numerical examples.

Local improvements to reduced-order approximations of optimal control problems governed by diffusion–convection–reaction equation

We consider the optimal control problem governed by diffusion–convection–reaction equation without control constraints. The proper orthogonal decomposition (POD) method is used to reduce the dimension of the problem. The POD method may lack accuracy if the POD basis depending on a set of parameters is used to approximate the problem depending on a different set of parameters. To increase the accuracy and the robustness of the basis, we compute five bases additional to the baseline POD in case of the perturbation of the diffusion term, a parameter in the convection field, the reaction term and Tikhonov regularization term. For the first two bases, we use the sensitivity information to extrapolate and expand the baseline POD basis. The other one is based on the subspace angle interpolation method. Multiple snapshot sets are used to derive the last two bases. A-posteriori error estimator is used to analyse the difference between the suboptimal control, computed using the POD basis, and the optimal control. We compare these different bases in terms of accuracy and complexity, investigate the advantages and main drawbacks of them.

A Goal-Oriented Error Estimator for Reduced Basis Method Modeling of Microwave Devices

This letter proposes a novel a-posteriori error estimator suitable for the reduced order modeling of microwave circuits. Unlike the existing error estimators based on impedance function residuals, the new one exploits the residual error associated with the computation of the scattering matrix. The estimator can be effectively used in the Reduced Basis Method (RBM) to automatically generate reduced-order models. The results of numerical experiments show that the new error estimator predicts the errors in the computed S-parameters more accurately than the other existing approaches, and that it can be used to automate the generation of wideband reduced models.

Error Control for the Localized Reduced Basis Multiscale Method with Adaptive On-Line Enrichment

In this contribution we consider localized, robust and efficient a-posteriori error estimation of the localized reduced basis multi-scale (LRBMS) method for parametric elliptic problems with possibly heterogeneous diffusion coefficient. The numerical treatment of such parametric multi-scale problems are characterized by a high computational complexity, arising from the multi-scale character of the underlying differential equation and the additional parameter dependence. The LRBMS method can be seen as a combination of numerical multi-scale methods and model reduction using reduced basis (RB) methods to efficiently reduce the computational complexity with respect to the multi-scale as well as the parametric aspect of the problem, simultaneously. In contrast to the classical residual based error estimators currently used in RB methods, we are considering error estimators that are based on conservative flux reconstruction and provide an efficient and rigorous bound on the full error with respect to the weak solution. In addition, the resulting error estimator is localized and can thus be used in the on-line phase to adaptively enrich the solution space locally where needed. The resulting certified LRBMS method with adaptive on-line enrichment thus guarantees the quality of the reduced solution during the on-line phase, given any (possibly insufficient) reduced basis that was generated during the offline phase. Numerical experiments are given to demonstrate the applicability of the resulting algorithm with online enrichment to single phase flow in heterogeneous media.

A class of iterative methods for solving nonlinear operator equations

The article deals with gradient-like iterative methods for solving nonlinear operator equations on Hilbert and Banach spaces. The authors formulate a general principle of studying such methods. This principle allows to formulate simple conditions of convergence of the method under consideration, to estimate the rate of this convergence and to give effective APRIORI and APOSTERIORI error estimates in terms of a scalar function that is constructed on the base of estimates for properties of invertibility and smoothness of linearizations of the left-hand side of the equations under study. The principle is applicable for analysis of such classical methods as method of minimal residuals, method of steepest descent, method of minimal errors and others. The main results are obtained for operator equations on Hilbert spaces and Banach spaces with a special property, that is called Bynum property.

A comparison of duality and energy a posteriori estimates for $\mathrm {L}_{\infty }(0,T;\mathrm {L}_2(\varOmega ))$ in parabolic problems

We use the elliptic reconstruction technique in combination with a duality approach to prove aposteriori error estimates for fully discrete back- ward Euler scheme for linear parabolic equations. As an application, we com- bine our result with the residual based estimators from the aposteriori esti- mation for elliptic problems to derive space-error indicators and thus a fully practical version of the estimators bounding the error in the L \infty (0, T ; L2({\Omega})) norm. These estimators, which are of optimal order, extend those introduced by Eriksson and Johnson (1991) by taking into account the error induced by the mesh changes and allowing for a more flexible use of the elliptic estima- tors. For comparison with previous results we derive also an energy-based aposteriori estimate for the L \infty (0, T ; L2({\Omega}))-error which simplifies a previous one given in Lakkis and Makridakis (2006). We then compare both estimators (duality vs. energy) in practical situations and draw conclusions.

Reduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system

The coupling of a free flow with a flow through porous media has many potential applications in several fields related with computational science and engineering, such as blood flows, environmental...

Transient response analysis of randomly parametrized finite element systems based on approximate balanced reduction

A model order reduction scheme of the transient response of large-scale randomly parametrized linear finite element system in state space form has been proposed. The reduced order model realization is aimed at preserving the invariant properties of the dynamic system model based on the dominant coupling characteristics of the specified system inputs and outputs. An a-priori model reduction strategy based on the balanced truncation method has been proposed in conjunction with the stochastic spectral Galerkin finite element method. Approximation of the dominant modes of the controllability Gramian has been performed with iterative Arnoldi scheme applied to Lyapunov equations. The reduced order representation of the randomly parametrized dynamical system has been obtained with Arnoldi–Lyapunov vector basis using an implicit time stepping algorithm. The performance and the computational efficacy of the proposed scheme has been illustrated with examples of randomly parametrized advection–diffusion–reaction problem under the action of transient external forcing functions. The convergence of the proposed reduced order scheme has been shown with a-posteriori error estimates.

Certified PDE-constrained parameter optimization using reduced basis surrogate models for evolution problems

We consider parameter optimization problems which are subject to constraints given by parametrized partial differential equations. Discretizing this problem may lead to a large-scale optimization problem which can hardly be solved rapidly. In order to accelerate the process of parameter optimization we will use a reduced basis surrogate model for numerical optimization. For many optimization methods sensitivity information about the functional is needed. In the following we will show that this derivative information can be calculated efficiently in the reduced basis framework in the case of a general linear output functional and parametrized evolution problems with linear parameter separable operators. By calculating the sensitivity information directly instead of applying the more widely used adjoint approach we can rapidly optimize different cost functionals using the same reduced basis model. Furthermore, we will derive rigorous a-posteriori error estimators for the solution, the gradient and the optimal parameters, which can all be computed online. The method will be applied to two parameter optimization problems with an underlying advection-diffusion equation.

Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems

In this work we introduce and analyse a new adaptive Petrov-Galerkin heterogeneous multiscale finiteelement method (HMM) for monotone elliptic operators with rapid oscillations.In a general heterogeneous setting we prove convergence of theHMM approximations to the solution of a macroscopic limit equation.The major new contribution of this work is an a-posteriori error estimatefor the $L^2$-error between the HMM approximation and the solution of themacroscopic limit equation.The a posteriori error estimate is obtained in a general heterogeneous settingwith scale separation without assuming periodicity or stochastic ergodicity.The applicability of the method and the usage of the a posteriori error estimatefor adaptive local mesh refinement is demonstrated in numerical experiments.The experimental results underline the applicability of the a posteriori errorestimate in non-periodic homogenization settings.

A-posteriori error analysis for a staggered discontinuous Galerkin discretization of the time-harmonic Maxwell’s equations

In this paper, we present the first a-posteriori error analysis for the staggered discontinuous Galerkin (SDG) method. Specifically, we consider the approximation of the time-harmonic Maxwell’s equations by a SDG method, and prove that our residual based a-posteriori error indicator is both reliable and efficient. We validate the performance of the indicator within an adaptive mesh refinement procedure and show its asymptotic exactness for a range of test problems.