Posteriori Error Estimates Research Articles

A review of VMS a posteriori error estimation with emphasis in fluid mechanics

This article outlines the research on the application of the variational multiscale theory (VMS) to a posteriori error estimation. VMS theory was initially developed by Professor Hughes to evince the origins of stabilized methods. In this paper it is shown that the stabilization parameters and the stabilization terms contain true error information that can be used to obtain explicit and implicit a posteriori error estimates. The technology consists of splitting the exact solution into resolved or coarse scales (finite element solution) and unresolved or fine scales (numerical error). By feeding this splitting into the variational formulation, an exact weak form can be derived for the fine scales as a function of the resolved scales. The way of solving or approximating this equation yields different algorithms and models for error estimation. Furthermore, using the so-called fine-scale Green’s function, an analytical representation of the fine scales is possible. Again, different approximations of this function give rise to various algorithms and models. This theory naturally suggests that the error can be computed by the combination of element interior and inter-element faces residuals with the corresponding error time-scales. From this standpoint, error estimators are developed for the transport equation and the Navier–Stokes equations. This technology can be further used for example to generate adapted meshes, to derive reduced order models and in verification and validation algorithms.

An augmented mixed FEM for the convective Brinkman–Forchheimer problem: a priori and a posteriori error analysis

We propose and analyze an augmented mixed finite element method for the pseudostress–velocity formulation of the stationary convective Brinkman–Forchheimer problem in Rd,d∈{2,3}. Since the convective and Forchheimer terms forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax–Milgram theorem, allow to prove the unique solvability of the continuous problem. The finite element discretization involves Raviart–Thomas spaces of order k≥0 for the pseudostress tensor and continuous piecewise polynomials of degree ≤k+1 for the velocity. Stability, convergence, and a priori error estimates for the associated Galerkin scheme are obtained. In addition, we derive two reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity of the form involved, a suitable assumption on the data, a stable Helmholtz decomposition, and the local approximation properties of the Clément and Raviart–Thomas operators. In turn, inverse inequalities, the localization technique based on bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, some numerical examples illustrating the performance of the mixed finite element method, confirming the theoretical rate of convergence and the properties of the estimators, and showing the behavior of the associated adaptive algorithms, are reported. In particular, the case of flow through a 2D porous media with fracture networks is considered.

Assessing the Structural Performance of Biodegradable Capsules

Biodegradable materials pose challenges over all aspects of computational mechanics. In this study, the focus is on the resulting domain uncertainty. Model structures or devices are shells of revolution subject to random variation of the outer surface. The novelty of the proposed computational approach is the possibility to restrict the variation to specific parts of the structure using a posteriori filtering, which is applied to the random process whose realisations are the profiles of the shells. The dimensionally reduced stochastic elasticity problems are solved using a collocation method where every realisation is discretised separately. The collocation scheme is validated against standard Monte Carlo. The reliability of the simulations is further confirmed via a posteriori error estimates that are computed using the same collocation scheme. The quantities of interest on the nominal domain are the expected displacement fields and their variances.

An adaptive finite element method for Riesz fractional partial integro-differential equations

AbstractThe Riesz fractional derivative has been employed to describe the spatial derivative in a variety of mathematical models. In this work, the accuracy of the finite element method (FEM) approximations to Riesz fractional derivative was enhanced by using adaptive refinement. This was accomplished by deducing the Riesz derivatives of the FEM bases to work on non-uniform meshes. We utilized these derivatives to recover the gradient in a space fractional partial integro-differential equation in the Riesz sense. The recovered gradient was used as an a posteriori error estimator to control the adaptive refinement scheme. The stability and the error estimate for the proposed scheme are introduced. The results of some numerical examples that we carried out illustrate the improvement in the performance of the adaptive technique.

THE GREEN-RVACHOV QUASI-FUNCTION METHOD IN THE NUMERICAL ANALYSIS OF MICROELECTROMECHANICAL SYSTEMS USING TWO-SIDED APPROXIMATIONS

The work is devoted to developing a bilateral iterative method of numerical analysis of one electrostatic microelectromechanical system based on the use of the Green – Rvachev quasi-function. Microelectromechanical systems are miniature devices that combine electronic and mechanical components of micron size. Electrostatically activated microelectromechanical systems have certain disadvantages that limit the range of their operation. One of these is the pull-in instability of the functional components of the system, which occurs if the applied voltage difference is above a particular critical value. The mathematical model of the system under consideration in the work is a semi-linear elliptic equation with the Laplace operator and the Dirichlet condition. To construct an approximate solution of the problem, it is proposed to use the methods of nonlinear analysis in semi-ordered spaces, particularly the results of V. I. Opoitsev on the solvability of nonlinear operator equations with a heterotonic operator. The boundary value problem simulating the most straightforward microelectromechanical system under the action of external pressure is reduced to the Urison integral equation using the Green – Rvachev quasi-function method, which makes it possible to expand the application of the method of two-sided approximations for problems in areas of somewhat arbitrary geometry. The paper substantiates the possibility of constructing iterative sequences with two-sided convergence to a positive solution of the problem: the computational scheme is given, the conditions for its convergence to the desired solution are obtained, and the a posteriori error estimate is derived. The method is illustrated by a computational experiment for a problem considered in a rectangular area. The computational experiment results are presented in the form of the surface and lines of the level of the approximate solution, and the two-sided nature of the convergence of the proposed method is also graphically illustrated.

Stabilization-free HHO a posteriori error control

The known a posteriori error analysis of hybrid high-order methods treats the stabilization contribution as part of the error and as part of the error estimator for an efficient and reliable error control. This paper circumvents the stabilization contribution on simplicial meshes and arrives at a stabilization-free error analysis with an explicit residual-based a posteriori error estimator for adaptive mesh-refining as well as an equilibrium-based guaranteed upper error bound (GUB). Numerical evidence in a Poisson model problem supports that the GUB leads to realistic upper bounds for the displacement error in the piecewise energy norm. The adaptive mesh-refining algorithm associated to the explicit residual-based a posteriori error estimator recovers the optimal convergence rates in computational benchmarks.

Posteriori error neural network: A recovery type posteriori error estimator based on neural network for diffusion problems

In this article, a posteriori error neural network (PENN) is proposed for the finite element approximate solution of diffusion problems. The PENN is constructed to solve a minimum problem which is obtained from the Prager and Syge equality. Then the output of the PENN can be used to estimate the posteriori error of the finite element approximate solution. There are two-fold differences compared to the classic methods. Firstly, the PENN provides a mesh-free recovery type error estimator. A flux is constructed by a neural network which does not need any given mesh. Secondly, once finishing the training procedure for an given approximate solution, the PENN can be used to estimate posteriori error for other approximate solution without training the PENN again. This feature is useful when we just use the posteriori error distribution as the indicator for adaptive refinement. The effectiveness of the PENN is demonstrated through a set of classical numerical experiments.

Adaptive interior penalty hybridized discontinuous Galerkin methods for Darcy flow in fractured porous media

Abstract In this paper, we design and analyze an interior penalty hybridized discontinuous Galerkin (IP-HDG) method for the Darcy flow in the two- and three-dimensional fractured porous media. The discrete fracture model is used to model the fractures. The piecewise polynomials of degree $k$ are employed to approximate the pressure in the fractures and the pressure in the surrounding porous media. We prove that the IP-HDG method is well posed if the penalty parameter is large enough. Based on the discrete solutions of pressures, the discrete Darcy velocity in the matrix and the reduced fractures can be recovered, respectively, to be locally mass-conservative. A robust residual-based a posteriori error estimator is established for an energy-norm of pressure. Finally, numerical results are provided to show the efficiency of the proposed a posteriori error estimator.

An overview of a posteriori error estimation and post-processing methods for nonlinear eigenvalue problems

In this article, we present an overview of different a posteriori error analysis and post-processing methods proposed in the context of nonlinear eigenvalue problems, e.g. arising in electronic structure calculations for the calculation of the ground state and compare them. We provide two equivalent error reconstructions based either on a second-order Taylor expansion of the minimized energy, or a first-order expansion of the nonlinear eigenvalue equation. We then show how several a posteriori error estimations as well as post-processing methods can be formulated as specific applications of the derived reconstructed errors, and we compare their range of applicability as well as numerical cost and precision.

A posteriori error estimates for wave maps into spheres

We provide a posteriori error estimates in the energy norm for temporal semi-discretisations of wave maps into spheres that are based on the angular momentum formulation. Our analysis is based on novel weak–strong stability estimates which we combine with suitable reconstructions of the numerical solution. We present time-adaptive numerical simulations based on the a posteriori error estimators for solutions involving blow-up.

First-order system least-squares finite element method for singularly perturbed Darcy equations

We define and analyse a least-squares finite element method for a first-order reformulation of a scaled Brinkman model of fluid flow through porous media. We introduce a pseudostress variable that allows to eliminate the pressure variable from the system. It can be recovered by a simple post-processing. It is shown that the least-squares functional is uniformly equivalent, i.e., independent of the singular perturbation parameter, to a parameter dependent norm. This norm equivalence implies that the least-squares functional evaluated in the discrete solution provides an efficient and reliable a posteriori error estimator. Numerical experiments are presented.

A posteriori error estimation and adaptivity for multiple-network poroelasticity

The multiple-network poroelasticity (MPET) equations describe deformation and pressures in an elastic medium permeated by interacting fluid networks. In this paper, we (i) place these equations in the theoretical context of coupled elliptic–parabolic problems, (ii) use this context to derive residual-based a posteriori error estimates and indicators for fully discrete MPET solutions and (iii) evaluate the performance of these error estimators in adaptive algorithms for a set of test cases: ranging from synthetic scenarios to physiologically realistic simulations of brain mechanics.

A posteriori error estimates for the Large Eddy Simulation applied to incompressible fluids

We study the two dimensional time dependent Large Eddy Simulation method applied to the incompressible Navier–Stokes system with Smagorinsky’s eddy viscosity model and a filter width that depends on the local mesh size. The discrete model is based on the implicit Euler scheme and a conforming finite element method for the time and space discretizations, respectively. We establish a reliable and efficient a posteriori error estimation between the numerical LES solution and the exact solution of the original Navier–Stokes system, which involves three types of error indicators respectively related to the filter and to the discretizations in time and space. Numerical results show the effectiveness of adaptive simulations

A posteriori error estimate for a modified weak Galerkin method of 2D (curl) elliptic problems

A posteriori error estimate for a modified weak Galerkin method of 2D (curl) elliptic problems

The finite element method for modified transmission eigenvalues for inverse scattering in a fluid–solid interaction problem

Based on the work of Monk and Selgas (2022), we explore the finite element method for the modified transmission eigenvalues for inverse scattering in a fluid–solid interaction problem. Under minimal regularity on the solution, we give a complete error analysis including the a priori and the a posteriori error estimates. We also present adaptive computations.

Parametric estimation of a stability factor for certified reduced basis methods via adaptive Gaussian process

The a posteriori error analysis of a reduced basis (RB) method may certify the accuracy of an RB solution without the truth finite element (FE) solution. However, the error theory necessitates a numerically expensive stability factor such as coercivity or an inf–sup factor. Consequently, we perforce resort to the successive constraint method (SCM) to achieve a stability lower bound for a rigorous albeit rapid RB analysis. As an alternative, we propose an adaptive Gaussian process (AGP) to produce a stability lower bound for the rigorous, rapid, and sharp estimation of an RB solution error. We utilize linear elastostatic and Helmholtz acoustic problems with geometric parameterization to compare the AGP with the SCM for a numerical investigation. The results show that the AGP significantly surpasses the SCM. This is because the a posteriori error estimation of an RB solution obtained with the AGP is sharper and faster than that obtained with the SCM by factors of at least 103 and 10, respectively. With regard to a rigorous error estimation, the AGP succeeds for the linear elastostatic problem but fails for the Helmholtz acoustic problem when an RB size is one at parameter values incurring only resonance. Otherwise, it manages to attain a rigorous error estimation. To conclude, the AGP is observed to be heuristically more practical than the SCM for a certified RB analysis. This research may considerably aid a static condensation RB element (scRBE) method because the technique numerously repeats the a posteriori error estimation of a reduced bubble function for the a posteriori error estimation of an scRBE solution.

Electromagnetic Modeling Using Adaptive Grids – Error Estimation and Geometry Representation

AbstractThis review paper addresses the development of numerical modeling of electromagnetic fields in geophysics with a focus on recent finite element simulation. It discusses ways of estimating errors of our solutions for a perfectly matched modeling domain and the problems that arise from its insufficient representation. After a brief outline of early methods and modeling approaches, the paper mainly discusses the capabilities of the finite element method formulated on unstructured grids and the advantages of local h-refinement allowing for both a flexible and largely accurate representation of the geometries of the multi-scale geomaterial and an accurate evaluation of the underlying functions representing the physical fields. In summary, the accuracy of the solution depends on the geometric mapping, the choice of the mathematical model, and the spatial discretization. Although the available error estimators do not necessarily provide reliable error bounds for our complex geomodels, they are still useful to guide grid refinement. Therefore, an overview of the most common a posteriori error estimators is given. It will be shown that the sensitivity is the most important function in both guiding the geometric mapping and the local refinement.

Space−time spectral approximations of a parabolic optimal control problem with an L2‐norm control constraint

AbstractThis article deals with the spectral approximation of an optimal control problem governed by a parabolic partial differential equation (PDE) with an ‐norm control constraint. The investigations employ the space−time spectral method, which is, more precisely, a dual Petrov‐Galerkin spectral method in time and a spectral method in space to discrete the continuous system. As a global method, it uses the global polynomials as the trial functions for discretization of PDEs. After obtaining the optimality condition of the continuous system and that of its spectral discrete surrogate, we establish a priori and a posteriori error estimates for the spectral approximation in detail. Three numerical examples in different spatial dimensions then confirm the theoretical results and also show the efficiency as well as a good precision of the adopted space−time spectral method.

A simple equilibration procedure leading to polynomial-degree-robust a posteriori error estimators for the curl-curl problem

We introduce two a posteriori error estimators for Nédélec finite element discretizations of the curl-curl problem. These estimators pertain to a new Prager–Synge identity and an associated equilibration procedure. They are reliable and efficient, and the error estimates are polynomial-degree-robust. In addition, when the domain is convex, the reliability constants are fully computable. The proposed error estimators are also cheap and easy to implement, as they are computed by solving divergence-constrained minimization problems over edge patches. Numerical examples highlight our key findings, and show that both estimators are suited to drive adaptive refinement algorithms. Besides, these examples seem to indicate that guaranteed upper bounds can be achieved even in non-convex domains.

A posteriori error estimator for mixed interior penalty discontinuous Galerkin finite element method for the [formula omitted](curl)-elliptic problems

In this paper, we design the first residual type a posteriori error estimator for mixed interior penalty discontinuous Galerkin method for the H(curl)-elliptic problems. Then we prove that our indicator is both reliable and efficient. At last, we present some numerical experiments to validate the performance of the indicator within an adaptive mesh refinement procedure.