Posteriori Error Estimates Research Articles

An Equilibrated Flux A Posteriori Error Estimator for Defeaturing Problems

An Equilibrated Flux A Posteriori Error Estimator for Defeaturing Problems

On a posteriori error estimation for Runge–Kutta discontinuous Galerkin methods for linear hyperbolic problems

AbstractA posteriori bounds for the error measured in various norms for a standard second‐order explicit‐in‐time Runge–Kutta discontinuous Galerkin (RKDG) discretization of a one‐dimensional (in space) linear transport problem are derived. The proof is based on a novel space‐time polynomial reconstruction, hinging on high‐order temporal reconstructions for continuous and discontinuous Galerkin time‐stepping methods. Of particular interest is the question of error estimation under dynamic mesh modification. The theoretical findings are tested by numerical experiments.

A nonlinear least-squares convexity enforcing 𝐶⁰ interior penalty method for the Monge–Ampère equation on strictly convex smooth planar domains

We construct a nonlinear least-squares finite element method for computing the smooth convex solutions of the Dirichlet boundary value problem of the Monge-Ampère equation on strictly convex smooth domains in R 2 {\mathbb {R}}^2 . It is based on an isoparametric C 0 C^0 finite element space with exotic degrees of freedom that can enforce the convexity of the approximate solutions. A priori and a posteriori error estimates together with corroborating numerical results are presented.

A priori and a posteriori error analysis of a mixed DG method for the three-field quasi-Newtonian Stokes flow

Abstract In this paper we propose and analyse a new mixed-type DG method for the three-field quasi-Newtonian Stokes flow. The scheme is based on the introduction of the stress and strain tensor as further unknowns as well as the elimination of the pressure variable by means of the incompressibility constraint. As such, the resulting system involves three unknowns: the stress, the strain tensor and the velocity. All these three unknowns are approximated using discontinuous piecewise polynomials, which offers flexibility for enforcing the symmetry of the stress and the strain tensor. The unique solvability and a comprehensive convergence error analysis for all the variables are performed. All the variables are proved to converge optimally. Adaptive mesh refinement guided by a posteriori error estimator is computationally efficient, especially for problems involving singularity. In line of this mechanism we derive a residual-type a posteriori error estimator, which constitutes the second main contribution of the paper. In particular, we employ the elliptic reconstruction in conjunction with the Helmholtz decomposition to derive the a posteriori error estimator, which avoids using the averaging operator. Several numerical experiments are carried out to verify the theoretical findings.

A priori and a posteriori error estimates for efficient numerical schemes for coupled systems of linear and nonlinear singularly perturbed initial-value problems

This work considers the numerical approximation of linear and nonlinear singularly perturbed initial value coupled systems of first-order, for which the diffusion parameters at each equation of the system are distinct and also they can have a different order of magnitude. To do that, we use two efficient discretization methods, which combine the backward differences and an appropriate splitting by components. Both a priori and a posteriori error estimates are proved for the proposed discretization methods. The developed numerical methods are more computationally efficient than those classical methods used to solve the same type of coupled systems. Extensive numerical experiments strongly confirm in practice the theoretical results and corroborate the superior performance of the current approach compared with previous existing approaches.

Дослідження процесів самозаймання у насипі з прямокутним перерізом методами Роте та двобічних наближень

Self-ignition of a stockpile of material (coal, peat, grain) occurs as a result of the accumulation of heat released by an exothermic reaction of oxidation, which makes it possible to consider the stockpile as a body with an internal heat source. The research of self-ignition processes using mathematical modeling methods leads to the need to find a solution to the initial boundary value problem for a two-dimensional semilinear heat conduction equation. This cannot always be done analytically, so it makes sense to use numerical analysis methods. The aim of this article is to apply Rothe’s method in combination with the method of two-sided approximations based on the use of the Green's function to find the solution of the initial boundary value problem for the two-dimensional semilinear heat conduction equation that arises during the mathematical modeling of self-ignition processes of an stockpile of bulk material of cylindrical shape with a rectangular base. To achieve the goal, after the discretization of the heat conduction equation by the time variable, a sequence of boundary value problems was obtained by the Rothe’s method, each of which was reduced to the Hammerstein equation. For this nonlinear operator equation, an iterative process of the two-way method was constructed with its stopping condition obtained through a posteriori error estimation. The power of the internal heat source was approximated using an exponential dependence. As a result of the computational experiment, a sequence of approximate solutions was obtained. The graphs of heat maps constructed for them made it possible to examine over time the course of the self-ignition process in the cross-section of a stockpile of cylindrical shape with a rectangular base and to identify areas of heat accumulation

Hp-FEM for the α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Mosolov problem: a priori and a posteriori error estimates

An hp-finite element discretization for the α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Mosolov problem, a scalar variant of the Bingham flow problem but with the α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Laplacian operator, is being analyzed. Its weak formulation is either a variational inequality of second kind or equivalently a non-smooth but convex minimization problem. For any α∈(1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (1,\\infty )$$\\end{document} we prove convergence, including guaranteed convergence rates in the mesh size h and polynomial degree p of the FE-solution of the corresponding discrete variational inequality. Moreover, we derive two families of reliable a posteriori error estimators which are applicable to any “approximation” of the exact solution and not only to the FE-solution and can therefore be coupled with an iterative solver. We prove that any quasi-minimizer of those families of a posteriori error estimators satisfies an efficiency estimate. All our results contain known results for the Mosolov problem by setting α=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =2$$\\end{document}. Numerical results underline our theoretical findings.

Stability of step size control based on a posteriori error estimates

A posteriori error estimates based on residuals can be used for reliable error control of numerical methods. Here, we consider them in the context of ordinary differential equations and Runge-Kutta methods. In particular, we take the approach of Dedner & Giesselmann (2016) and investigate it when used to select the time step size. We focus on step size control stability when combined with explicit Runge-Kutta methods and demonstrate that a standard I controller is unstable while more advanced PI and PID controllers can be designed to be stable. We compare the stability properties of residual-based estimators and classical error estimators based on an embedded Runge-Kutta method both analytically and in numerical experiments.

Regularization method and a posteriori error estimates for the bilateral obstacle problem

In this paper, we present a relatively complete analysis of the regularization method for of the bilateral obstacle problem with a non-homogeneous condition on the boundary. We provide both the convergence result and a priori estimates. Then by the duality theory from the convex analysis, we determine the dual problem associated with the of the bilateral obstacle problem. Based on this dual problem, we provide a posteriori error estimates for the continuous and the discrete problems. The a posteriori error estimates are desired for the practical implementation of the regularized problem.

Reduced basis method for non-symmetric eigenvalue problems: application to the multigroup neutron diffusion equations

In this article, we propose a reduced basis method for parametrized non-symmetric eigenvalue problems arising in the loading pattern optimization of a nuclear core in neutronics. To this end, we derive a posteriori error estimates for the smallest eigenvalue which is assumed to be simple and the associated left and right eigenvectors. The practical computation of these estimators requires the estimation of a constant called prefactor, which we can express as the spectral norm of some operator. We provide some elements of theoretical analysis which illustrate the link between the expression of the prefactor we obtain here and its well-known expression in the case of symmetric eigenvalue problems, either using the notion of numerical range of the operator, or via a perturbative analysis. Lastly, we propose a practical method in order to estimate this prefactor which yields interesting numerical results on actual test cases. We provide detailed numerical simulations on two-dimensional examples including a multigroup neutron diffusion equation.

An arbitrarily high order unfitted finite element method for elliptic interface problems with automatic mesh generation, Part II. Piecewise-smooth interfaces

We consider the reliable implementation of an adaptive high-order unfitted finite element method on Cartesian meshes for solving elliptic interface problems with geometrically curved singularities. We extend our previous work on the reliable cell merging algorithm for smooth interfaces to automatically generate the induced mesh for piecewise smooth interfaces. An hp a posteriori error estimate is derived for a new unfitted finite element method whose finite element functions are conforming in each subdomain. Numerical examples illustrate the competitive performance of the method.

Error Estimates and Adaptivity of the Space-Time Discontinuous Galerkin Method for Solving the Richards Equation

We present a higher-order space-time adaptive method for the numerical solution of the Richards equation that describes a flow motion through variably saturated media. The discretization is based on the space-time discontinuous Galerkin method, which provides high stability and accuracy and can naturally handle varying meshes. We derive reliable and efficient a posteriori error estimates in the residual-based norm. The estimates use well-balanced spatial and temporal flux reconstructions which are constructed locally over space-time elements or space-time patches. The accuracy of the estimates is verified by numerical experiments. Moreover, we develop the hp-adaptive method and demonstrate its efficiency and usefulness on a practically relevant example.

A posteriori-based, local multilevel mesh refinement for the Darcy porous media flow problem

In this work we develop an a posteriori-based adaptive local multilevel mesh refinement strategy for the Darcy porous media flow problem discretized with the finite volume method. Our approach is based on coupling the FIC “flux interface correction” method [1] and the equilibrated flux a posteriori error estimate [2]. The choice of the FIC method is motivated by the fact that it is well-adapted to problems discretized in conservative form since that the FIC provides corrections by balancing fluxes computed from both coarse and fin grids across the interface. The equilibrated flux a posteriori error estimate provides a guaranteed upper bound on the total error in the fluxes and takes advantage of the conservative scheme in such a way that the estimate can be easily coded and cheaply evaluated. We use this a posteriori estimate to detect automatically the zone of interest where the error is dominant in order to defined the local sub-grids. Numerous numerical experiments on practical problems illustrate the performance of our methodology. A comparison with a classical h-adaptive method is also carried out in this work.

Finite element approximation of unique continuation of functions with finite dimensional trace

We consider a unique continuation problem where the Dirichlet trace of the solution is known to have finite dimension. We prove Lipschitz stability of the unique continuation problem and design a finite element method that exploits the finite dimensionality to enhance stability. Optimal a priori and a posteriori error estimates are shown for the method. The extension to problems where the trace is not in a finite dimensional space, but can be approximated to high accuracy using finite dimensional functions is discussed. Finally, the theory is illustrated in some numerical examples.

Mixed finite elements for Kirchhoff–Love plate bending

We present a mixed finite element method with triangular and parallelogram meshes for the Kirchhoff–Love plate bending model. Critical ingredient is the construction of low-dimensional local spaces and appropriate degrees of freedom that provide conformity in terms of a sufficiently large tensor space and that allow for any kind of physically relevant Dirichlet and Neumann boundary conditions. For Dirichlet boundary conditions and polygonal plates, we prove quasi-optimal convergence of the mixed scheme. An a posteriori error estimator is derived for the special case of the biharmonic problem. Numerical results for regular and singular examples illustrate our findings. They confirm expected convergence rates and exemplify the performance of an adaptive algorithm steered by our error estimator.

An exactly divergence-free hybridized discontinuous Galerkin method for the generalized Boussinesq equations with singular heat source

The purpose of this work is to propose and analyze a hybridized discontinuous Galerkin (HDG) method for the generalized Boussinesq equations with singular heat source. We use polynomials of order k, k − 1 and k to approximate the velocity, the pressure and the temperature. By introducing Lagrange multipliers for the pressure, the approximate velocity field obtained by the HDG method is shown to be exactly divergence-free and H(div)-conforming. Under a smallness assumption on the problem data and solutions, we prove by Brouwer’s fixed point theorem that the discrete system has a solution in two dimensions. If the viscosity and thermal conductivity are further assumed to be positive constants, a priori error estimates with convergence rate O(h) and efficient and reliable a posteriori error estimates are derived. Finally numerical examples illustrate the theoretical analysis and show the performance of the obtained a posteriori error estimator.

An adaptive grid method for a two-parameter singularly perturbed problem with non-smooth data

In this paper, a two-parameter singularly perturbed problem with discontinuous source and convection coefficient is studied. This problem is discretized by using a first-order upwind finite difference scheme for which a posteriori error analysis in the maximum norm is derived. Then, based on this a posteriori error estimation, a grid iteration algorithm is designed to generate an adaptive nonuniform mesh. Finally, Numerical experiments are proposed to confirm that our proposed method is first-order uniformly convergent.

Adaptive meshing strategies for nanophotonics using a posteriori error estimation

As nanophotonic devices become increasingly complex, computer simulations of such devices are becoming ever more important. Unfortunately, computer simulations of nanophotonic devices are computationally expensive, especially if many simulations are necessary, e.g., when optimizing or inverse designing a device. Here we study adaptive mesh refinement for finite-element method simulations using an a posteriori error estimation method. We demonstrate that the use of adaptive meshing leads to faster convergence with lower memory footprint for complex three-dimensional nanophotonic structures. Nevertheless, one needs to be careful to avoid a mesh propagation effect for adaptive mesh refinement to be a successful strategy.

Residual-based a posteriori error estimators for algebraic stabilizations

In this note, we extend the analysis for the residual-based a posteriori error estimators in the energy norm defined for the algebraic flux correction (AFC) schemes (Jha, 2021) to the newly proposed algebraic stabilization schemes (John and Knobloch, 2022; Knobloch, 2023). Numerical simulations on adaptively refined grids are performed in two dimensions showing the higher efficiency of an algebraic stabilization with similar accuracy compared with an AFC scheme.

Error Approximation of the Second Order Hyperbolic Differential Equationby Using DG Finite Element Method

This article presents a simple efficient and asynchronously correcting a posteriori error approximation for discontinuous finite element solutions of the second-order hyperbolic partial differential problems on triangular meshes. This study considersthe basis functions for error spaces corresponding to some finite element spaces. The discretization error of each triangle is estimated by solving the local error problem. It also shows global super convergence for discontinuous solution on triangular lattice. In this article, the triangular elements are classify into three types: (i) elements with one inflow and two outflow edges are of type I, (ii) elements with two inflows and one outflow edges are of type II and (iii) elements with one inflow edge, one outflow edge, and one edge parallel to the characteristics are of type III. The article investigated higher-dimension discontinuous Galerkin methods for hyperbolic problems on triangular meshes and also studied the effect of finite element spaces on the superconvergence properties of DG solutions on three types of triangular elements and it showed that the DG solution is <i>O(h<sup>p+2</sup>)</i> superconvergent at Legendre points on the outflow edge on triangles having one outflow edge using three polynomial spaces. A posteriori error estimates are tested on a number of linear and nonlinear problems to show their efficiency and accuracy under lattice refinement for smooth and discontinuous solutions.