Posteriori Error Estimates Research Articles

Modified Trapezoidal Product Cubature Rules: Definiteness, Monotonicity, and a Posteriori Error Estimates

We study two modifications of the trapezoidal product cubature formulae, approximating double integrals over the square domain [a,b]2=[a,b]×[a,b]. Our modified cubature formulae use mixed type data: except evaluations of the integrand on the points forming a uniform grid on [a,b]2, they involve two or four univariate integrals. A useful property of these cubature formulae is that they are definite of order (2,2), that is, they provide one-sided approximation to the double integral for real-valued integrands from the class C2,2[a,b]={f(x,y):∂4f∂x2∂y2continuousanddoesnotchangesignin(a,b)2}. For integrands from C2,2[a,b] we prove monotonicity of the remainders and derive a posteriori error estimates.

The a posteriori error estimates and an adaptive algorithm of the discontinuous Galerkin method for the modified transmission eigenvalue problem with absorbing media

The a posteriori error estimates and an adaptive algorithm of the discontinuous Galerkin method for the modified transmission eigenvalue problem with absorbing media

A divergence‐conforming method for flow and double‐diffusive transport

AbstractThe results of a recent extension of the analysis of an ‐conforming method for a model of double‐diffusive flow in porous media introduced in [Bürger, Méndez, Ruiz‐Baier, SINUM (2019), 57:1318–1343] to the time‐dependent case are summarized. These include the efficiency and reliability of residual‐based a posteriori error estimators for the steady, semi‐discrete, and fully discrete problems. The method consists of Brezzi–Douglas–Marini approximations for velocity and compatible piecewise discontinuous pressures, whereas Lagrangian elements are used for concentration and salinity. Novel numerical tests confirm the accuracy of the method and illustrate its application to a salinity‐driven problem of sedimentation.

A unified framework for the error analysis of physics-informed neural networks

Abstract We prove a priori and a posteriori error estimates for physics-informed neural networks (PINNs) for linear PDEs. We analyze elliptic equations in primal and mixed form, elasticity, parabolic, hyperbolic and Stokes equations, and a PDE constrained optimization problem. For the analysis, we propose an abstract framework in the common language of bilinear forms, and we show that coercivity and continuity lead to error estimates. The obtained estimates are sharp and reveal that the $L^{2}$ penalty approach for initial and boundary conditions in the PINN formulation weakens the norm of the error decay. Finally, utilizing recent advances in PINN optimization, we present numerical examples that illustrate the ability of the method to achieve accurate solutions.

An Equilibrated Flux A Posteriori Error Estimator for Defeaturing Problems

An Equilibrated Flux A Posteriori Error Estimator for Defeaturing Problems

Adaptive option pricing based on a posteriori error estimates for fully discrete finite difference methods

In this paper we study the a posteriori error estimates of numerical methods for parabolic partial integro-differential equations (PIDEs) which describe jump-diffusion option pricing models in finance. Due to the non-smoothness of the payoff function, the initial singularity of the solution may arise and therefore a posteriori error control and adaptivity are of the utmost importance in numerically solving such type of equations. The implicit-explicit (IMEX) Euler and IMEX Crank–Nicolson Adams–Bashforth methods are used for the time discretization and a second order finite difference method is used for the space discretization. By using the continuous and piecewise reconstructions, the upper and lower bounds of the a posteriori error estimates for the fully discrete finite difference methods are derived, and these error bounds depend on the discretization parameters and the data of the model problems only. Based on these a posteriori error estimates, we further develop a spatial–temporal adaptive algorithm. Numerical experiments are performed with uniform partitions and time-space adaptivity. The new adaptive algorithm significantly reduces the computational cost and provides effective error control for the numerical solutions.

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

An adaptive hybridizable discontinuous Galerkin method for Darcy–Forchheimer flow in fractured porous media

In this paper, we consider an adaptive hybridizable discontinuous Galerkin (HDG) method based on the discrete fracture model for approximation of a single-phase flow in fractured porous media. We are interested in the case that the flow rate in fracture is large enough to justify the use of Forchheimer’s law for modeling flow within the fracture, while Darcy’s law is applied to the surrounding matrix. The HDG method could be designed to simulate the flow in porous media with reduced fractures which consist of many straight lines or planes. More specifically, we use piecewise polynomials of degree [Formula: see text] to approximate the velocity and pressure in fracture and surrounding porous media. The existence and uniqueness of discrete solutions are proved by the Brouwer fixed point theorem, and an efficient and reliable a posteriori error estimator is obtained with respect to an energy norm. Moreover, the HDG scheme, the existence and uniqueness of discrete solutions, and the a posteriori error estimates are also extended to the problem with non-planar, embedded, and intersecting fractures. Finally, several numerical examples are provided to validate the performance of the obtained a posteriori error estimator.

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.