Posteriori Error Indicator Research Articles

Meshfree Variational-Physics-Informed Neural Networks (MF-VPINN): An Adaptive Training Strategy

In this paper, we introduce a Meshfree Variational-Physics-Informed Neural Network. It is a Variational-Physics-Informed Neural Network that does not require the generation of the triangulation of the entire domain and that can be trained with an adaptive set of test functions. In order to generate the test space, we exploit an a posteriori error indicator and add test functions only where the error is higher. Four training strategies are proposed and compared. Numerical results show that the accuracy is higher than the one of a Variational-Physics-Informed Neural Network trained with the same number of test functions but defined on a quasi-uniform mesh.

Towards chemical accuracy using a multi-mesh adaptive finite element method in all-electron density functional theory

Chemical accuracy serves as an important metric for assessing the effectiveness of the numerical method in Kohn–Sham density functional theory. It is found that to achieve chemical accuracy, not only the Kohn–Sham wavefunctions but also the Hartree potential, should be approximated accurately. Under the adaptive finite element framework, this can be implemented by constructing the a posteriori error indicator based on approximations of the aforementioned two quantities. However, this way results in a large amount of computational cost. To reduce the computational cost, we propose a novel solver for the Kohn–Sham equation by using the multi-mesh adaptive technique, in which the Kohn–Sham equation and the Poisson equation are solved in two different meshes on the same computational domain, respectively. With the proposed method, chemical accuracy can be achieved with less computational consumption compared with the adaptive method on a single mesh, as demonstrated in a number of numerical experiments.

Divergence-conforming methods for transient double-diffusive flows: a priori and a posteriori error analysis

Abstract The analysis of an $\textbf {H}(\textrm {div})$-conforming method for a model of double-diffusive flow in porous media introduced in Bürger, Méndez & Ruiz-Baier (2019, On H(div)-conforming methods for double-diffusion equations in porous media. SIAM J. Numer. Anal., 57,1318–1343) is extended to the time-dependent case. In addition, the efficiency and reliability of residual-based a posteriori error estimators for the steady, semidiscrete and fully discrete problems are established. The resulting methods are applied to simulate the sedimentation of small particles in salinity-driven flows. The method consists of Brezzi–Douglas–Marini approximations for velocity and compatible piecewise discontinuous pressures, whereas Lagrangian elements are used for concentration and salinity distribution. Numerical tests confirm the properties of the proposed family of schemes and of the adaptive strategy guided by the a posteriori error indicators.

A Mixed Finite Element Approximation for Time-Dependent Navier–Stokes Equations with a General Boundary Condition

In this paper, we present a numerical scheme for addressing the unsteady asymmetric flows governed by the incompressible Navier–Stokes equations under a general boundary condition. We utilized the Finite Element Method (FEM) for spatial discretization and the fully implicit Euler scheme for time discretization. In addition to the theoretical analysis of the error in our numerical scheme, we introduced two types of a posteriori error indicators: one for time discretization and another for spatial discretization, aimed at effectively controlling the error. We established the equivalence between these estimators and the actual error. Furthermore, we conducted numerical simulations in two dimensions to assess the accuracy and effectiveness of our scheme.

A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering

Abstract Numerical methods for computing Steklov eigenvalues have attracted the attention of academia for their important physical background and wide applications. In this article we discuss the multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering, and give the error estimation of the proposed scheme. In addition, on the basis of the a posteriori error indicator, we design an adaptive multigrid algorithm. Finally, we present numerical examples to show the efficiency of the proposed scheme.

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.

An improved model for the drying of porous solids with abruptly changing permeability

We present the simulation of the drying of porous solids with abruptly changing permeability using the Darcy-Forchheimer equations. This model arises in fluid mechanics through porous media when the flow exhibits moderate to high velocities and can be useful to predict the drying rate of the first drying period, which is usually constant. The presence of abrupt changes in permeability creates strong singularities and makes it difficult to obtain accurate predictions of the fluid flow, velocity, and pressure drop. In order to overcome these difficulties, we apply an adaptive finite element method based on a simple a posteriori error indicator, reliable and locally efficient, that allows us to refine the mesh over the singularities. We also present a semi-analytical method to obtain an accurate measurement of the apparent permeability of a porous medium. Finally, we provide some numerical experiments that illustrate the performance of the method.

A Posteriori Error Estimators for the Quasi-Newtonian Stokes Problem with a General Boundary Condition

In this paper, we approach two nonlinear differential equations applied in fluid mechanics by finite element methods (FEM). Our objective is to approach the solution to these problems; the first one is the “p-Laplacian” problem and the second one is the “Quasi-Newtonian Stokes” problem with a general boundary condition. To study and analyze our solutions, we introduce the a posteriori error indicator; this technique allows us to control the error, and each is shown the equivalent between the true and the a posterior errors estimators. The performance of the finite element method by this type of general boundary condition is presented via different numerical simulations.

A posteriori error estimates for Biot system using a mixed discretization for flow

We first derive convergence and a priori stability, and next reliability and efficiency of a posteriori error indicators for a Biot poroelastic model coupled with an elastic model in R3, solved by a continuous Galerkin scheme (CG) for the displacement and a mixed finite element scheme for the flow. The coupled system is decoupled by a fixed stress splitting algorithm. The numerical implementation of the residual based error indicators is simple, even for the mixed discretization, but at the expense of two suboptimal bounds. The scheme is tested on two benchmark problems via several numerical experiments.

Stabilized finite elements for Tresca friction problem

We formulate and analyze a Nitsche-type algorithm for frictional contact problems. The method is derived from, and analyzed as, a stabilized finite element method and shown to be quasi-optimal, as well as suitable as an adaptive scheme through an a posteriori error analysis. The a posteriori error indicators are validated in a numerical experiment.

Analysis of an unfitted mixed finite element method for a class of quasi-Newtonian Stokes flow

We propose and analyze an unfitted method for a dual-dual mixed formulation of a class of Stokes models with variable viscosity depending on the velocity gradient, in which the pseudoestress, the velocity and its gradient are the main unknowns. On a fluid domain Ω with curved boundary Γ we consider a Dirichlet boundary condition and employ an approach previously applied to the Stokes equations with constant viscosity, which consists of approximating Ω by a polyhedral computational subdomain Ωh, not necessarily fitting Ω, where a Galerkin method is applied to compute solution. Furthermore, to approximate the Dirichlet data on the computational boundary Γh, we make use of a transferring technique based on integrating the discrete velocity gradient. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas of order k≥0 for the pseudostress, and discontinuous polynomials of degree k for the velocity and its gradient. For the a priori error analysis we provide suitable assumptions on the mesh near the boundary Γ ensuring that the associated Galerkin scheme is well-posed and optimally convergent with O(hk+1). Next, for the case when Γh is taken as a piecewise linear interpolation of Γ, we develop a reliable and quasi-efficient residual-based a posteriori error estimator. Numerical experiments verify our analysis and illustrate the performance of the associated a posteriori error indicator.

A posteriori error analysis of a momentum conservative Banach spaces based mixed-FEM for the Navier–Stokes problem

In this paper we develop an a posteriori error analysis of a new momentum conservative mixed finite element method recently introduced for the steady-state Navier–Stokes problem in two and three dimensions. More precisely, by extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces, such as local estimates, and suitable Helmholtz decompositions, we derive a reliable and efficient residual-based a posteriori error estimator for the corresponding mixed finite element scheme on arbitrary (convex or non-convex) polygonal and polyhedral regions. On the other hand, inverse inequalities, the localization technique based on bubble functions, among other tools, are employed to prove the efficiency of the proposed a posteriori error indicator. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.

The a priori and a posteriori error estimates of Crouzeix–Raviart element for the fluid–solid vibration problem

In this paper, we study the nonconforming Crouzeix–Raviart (CR) element approximation for the fluid–solid vibration problem. We present the a priori error estimate and the a posteriori error estimate, and we prove the reliability and efficiency of the a posteriori error indicator up to higher order terms. Moreover, we carry out the numerical experiments in adaptive fashion which together with theoretical analysis show that the CR element eigenvalues approximate the exact ones from below.

Efficient numerical solution of one‐phase linear inverse Stefan and Cauchy–Stefan problems in two dimensions: A posteriori error control

AbstractThe efficient numerical solution of the one‐phase linear inverse Stefan and Cauchy–Stefan problems is a delicate task owing to the problems' susceptibility to the perturbation of the given data. In this context, heuristic a posteriori error indicators are constructed for such inverse problems with noisy data in two dimensions (2D). Given a fixed computational effort, the estimator controls the error due to discretization by the method of fundamental solution (MFS). It is accomplished through two mean‐driven double‐filtering algorithms. Numerical results substantiate the effectiveness of the algorithms.

A posteriori error estimates for stabilised mixed finite element methods for a nonlinear elliptic problem

In this paper we propose new adaptive stabilised mixed finite element methods for a nonlinear elliptic boundary value problem of second order in divergence form that appears, among other applications, in magnetostatics. The method is based on a three-field formulation that is augmented with suitable residual least-squares terms arising from the constitutive and equilibrium equations and from the equation that defines the gradient as an additional unknown. We show that the resulting scheme is well posed and obtain optimal error estimates. We also develop an a posteriori error analysis of residual type and derive a simple a posteriori error indicator which is reliable and locally efficient. Finally, we include several numerical experiments that confirm the theoretical results.

Efficient numerical solution of boundary identification problems: MFS with adaptive stochastic optimization

In this article, we study a novel computational technique for the efficient numerical solution of the inverse boundary identification problem with uncertain data in two dimensions. The method essentially relies on a posteriori error indicators consisting of the Tikhonov regularized solutions obtained by the method of fundamental solutions (MFS) and the given data for the problem in hand. For a desired accuracy, the a posteriori error estimator chooses the best possible combination of a complete set of fundamental solutions determined by the location of the sources that are arranged in a particular manner on a pseudo-boundary at each iteration. Also, since we are interested in a stable solution, an adaptive stochastic optimization strategy based on an error-balancing criterion is used, so as to avoid unstable regions where the stability contributions may be relatively large. These ideas are applied to two benchmark problems and are found to produce efficient and accurate results.

An adaptive strategy for discontinuous Galerkin simulations of Richards’ equation: Application to multi-materials dam wetting

Numerical solution of Richards’ equation remains challenging to get robust, accurate and cost-effective results, particularly for moving sharp wetting fronts. An adaptive strategy for both space and time is proposed to deal with 2D sharp wetting fronts associated with varying and possibly vanishing diffusivity caused by nonlinearity, heterogeneity and anisotropy. Adaptive time stepping makes nonlinear convergence reliable and backward difference formula provides high-order time scheme. Adaptive mesh refinement tracks wetting fronts with an a posteriori error indicator. The novelty of this paper consists in using this technique in combination with a weighted discontinuous Galerkin framework to better approximate steep wetting fronts by a discontinuity. The potential of the overall approach is shown through various examples including analytical and laboratory benchmarks and simulation of full-scale multi-materials dam wetting experiment.

Analysis of estimators for Stokes problem using a mixed approximation

In this work, we introduce the steady Stokes equations with a new boundary condition, generalizes the Dirichlet and the Neumann conditions. Then we derive an adequate variational formulation of Stokes equations. It includes algorithms for discretization by mixed finite element methods. We use a block diagonal preconditioners for Stokes problem. We obtain a faster convergence when applying the preconditioned MINRES method. Two types of a posteriori error indicator are introduced and are shown to give global error estimates that are equivalent to the true discretization error. In order to evaluate the performance of the method, the numerical results are compared with some previously published works or with others coming from commercial code like Adina system.

Error analysis of a conforming and locking-free four-field formulation for the stationary Biot’s model

We present ana priorianda posteriorierror analysis of a conforming finite element method for a four-field formulation of the steady-state Biot’s consolidation model. For thea priorierror analysis we provide suitable hypotheses on the corresponding finite dimensional subspaces ensuring that the associated Galerkin scheme is well-posed. We show that a suitable choice of subspaces is given by the Raviart–Thomas elements of orderk≥ 0 for the fluid flux, discontinuous polynomials of degreekfor the fluid pressure, and any stable pair of Stokes elements for the solid displacements and total pressure. Next, we develop a reliable and efficient residual-baseda posteriorierror estimator. Both the reliability and efficiency estimates are shown to be independent of the modulus of dilatation. Numerical examples in 2D and 3D verify our analysis and illustrate the performance of the proposeda posteriorierror indicator.

Resolution of Time-Dependent Navier-Stokes Equations with a new Boundary Condition

In this work, a numerical solution of the unsteady incompressible Navier- Stokes equations with a new boundary condition is proposed. The method suggested is based on an algorithm of discretization by finite element method in space and the Euler full-implicit scheme in time. The matrix system is solved at each iteration with a preconditioned GMRES method. Also, we proposed two types of a posteriori error indicator, with one being for the time discretization and the other for the space discretization. We prove the equivalence between the sum of the two types of error indicators and the full error. In order to evaluate the performance of the method, the numerical results of two-dimensional backward-facing step flow are compared with some previously published works or with others coming from commercial code like ADINA (Automatic Dynamic Incremental Nonlinear Analysis) system.