Error In Energy Norm Research Articles

Local discontinuous Galerkin method for a singularly perturbed fourth-order problem of convection-diffusion type

We develop a local discontinuous Galerkin (LDG) method for a fourth-order singularly perturbed problem of convection-diffusion type. The existence and uniqueness of the computed solution are verified. Using the Shishkin mesh we derive an optimal-order energy-norm error estimate which is uniformly valid in the perturbation parameter. Numerical experiments are also given to support our theoretical findings.

Energy norm error estimates and convergence analysis for a stabilized Maxwell’s equations in conductive media

The aim of this article is to investigate the well-posedness, stability of solutions to the time-dependent Maxwell’s equations for electric field in conductive media in continuous and discrete settings, and study convergence analysis of the employed numerical scheme. The situation we consider would represent a physical problem where a subdomain is emerged in a homogeneous medium, characterized by constant dielectric permittivity and conductivity functions. It is well known that in these homogeneous regions the solution to the Maxwell’s equations also solves the wave equation, which makes computations very efficient. In this way our problem can be considered as a coupling problem, for which we derive stability and convergence analysis. A number of numerical examples validate theoretical convergence rates of the proposed stabilized explicit finite element scheme.

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.

Balanced and energy norm error bounds for a spatial FEM with Crank-Nicolson and BDF2 time discretisation applied to a singularly perturbed reaction-diffusion problem

Balanced and energy norm error bounds for a spatial FEM with Crank-Nicolson and BDF2 time discretisation applied to a singularly perturbed reaction-diffusion problem

An inhomogeneous stabilized node-based smoothed radial point interpolation method for the multi-physics coupling responses of functionally graded magneto-electro-elastic structures

In this paper, an inhomogeneous stabilized node-based smoothed radial point interpolation method (ISNS-RPIM) is developed for analyzing functionally graded magneto-electro-elastic (FGMEE) structures in hygrothermal environment. The change of material gradient follows the exponential function distribution. Combining with the gradient smoothing technique (GST) and the coupling constitutive relationship, and considering the hygrothermo-magneto-electro-elastic (HMEE) coupling effects of FGMEE material, the equations for ISNS-RPIM are deduced. The generalized displacement of the FGMEE structures is obtained, and the error in total energy norm, CPU time and efficiency of ISNS-RPIM, NS-RPIM and FEM are researched. Numerical examples are provided to explore the influence of the following parameters in FGMEE structures including exponential factor, hygrothermal loading and empirical constants, and demonstrate the correctness, high effectivity and insensitivity to mesh distortion of ISNS-RPIM. The present method has shown great potential in solving practical complex problems.

Energy norm error estimate for singularly perturbed fourth-order differential equation with two parameters

We consider a fourth-order reaction–diffusion-type singularly perturbed boundary value problem with two small parameters and multiplied to the fourth- and second-order derivative terms respectively. In this article, we restrict to a special case, where and derive the finite element scheme using Ritz–Galerkin finite element method with lumping process. On discretizing the domain, the layer-adapted meshes like Standard-Shishkin, Bakhvalov-Shishkin and Modified-Bakhvalov-type meshes and piecewise quadratic polynomials are used and the error estimates are derived in -norm and energy norm. The numerical experiments given in the article supports these theoretical findings.

Fully discrete loosely coupled Robin-Robin scheme for incompressible fluid–structure interaction: stability and error analysis

We consider a fully discrete loosely coupled scheme for incompressible fluid–structure interaction based on the time semi-discrete splitting method introduced in Burman et al. (Numer Methods Partial Differ Equ, 2021). The splittling method uses a Robin-Robin type coupling that allows for a segregated solution of the solid and the fluid systems, without inner iterations. For the discretisation in space we consider piecewise affine continuous finite elements for all the fields and ensure the inf-sup condition by using a Brezzi-Pitkäranta type pressure stabilization. The interfacial fluid-stresses are evaluated in a variationally consistent fashion, that is shown to admit an equivalent Lagrange multiplier formulation. We prove that the method is unconditionally stable and robust with respect to the amount of added-mass in the system. Furthermore, we provide an error estimate that shows the error in the natural energy norm for the system is \({\mathcal {O}}\big (\sqrt{T}(\sqrt{\Delta t} + h)\big )\) where T is the final time, \(\Delta t\) the time-step length and h the space discretization parameter.

The local discontinuous Galerkin method on layer-adapted meshes for time-dependent singularly perturbed convection-diffusion problems

In this paper we analyze the error as well for the semi-discretization as the full discretization of a time-dependent convection-diffusion problem. We use for the discretization in space the local discontinuous Galerkin (LDG) method on a class of layer-adapted meshes including Shishkin-type and Bakhvalov-type meshes and the implicit θ-scheme in time. For piecewise tensor-product polynomials of degree k we obtain uniform or almost uniform error estimates with respect to space of order k+1/2 in some energy norm and optimal error estimates with respect to time. Our analysis is based on careful approximation error estimates for the Ritz projection related to the stationary problem on the anisotropic meshes used. We discuss also improved estimates in the one-dimensional case and the use of a discontinuous Galerkin discretization in time. Numerical experiments are given to support our theoretical results.

Further investigation of convolutional neural networks applied in computational electromagnetism under physics‐informed consideration

Convolutional neural networks (CNN) have shown great potentials and have been proven to be an effective tool for some image-based deep learning tasks in the field of computational electromagnetism (CEM). In this work, an energy-based physics-informed neural network (EPINN) is proposed for low-frequency electromagnetic computation. Two different physics-informed loss functions are designed. To help the network focus on the region of interest instead of computing the whole domain on average, the magnetic energy norm error loss function is proposed. Besides, the methodology of energy minimization is integrated into the CNN by introducing the magnetic energy error loss function. It is observed that the introduction of the physics-informed loss functions improved the accuracy of the network with the same architecture and database. Meanwhile, these changes also cause the network to be more sensitive to some hyperparameters and makes the training process oscillate or even diverge. To address this issue, the sensitivity of the network hyperparameters for both physics-informed loss functions are further investigated. Numerical experiments demonstrate that the proposed approaches have good accuracy and efficiency with fine-tuned hyperparameters. Furthermore, the post-test illustrates that the EPINN has excellent interpolation performance and can obtain good extrapolation results under certain restrictions.

Adaptive Isogeometric analysis for transient dynamics: Space–time refinement based on hierarchical a-posteriori error estimations

In this paper, based on our proposed IsoGeometric Analysis (IGA)-based adaptivity technique for plate vibration (Yu et al., 2018) the space–time adaptivity strategies used for transient dynamics are established. Specifically, the Geometry Independent Field approximaTion (GIFT) method is applied to discretize the spatial domain, and time discretization is proceeded by Newmark method. In the framework of GIFT/Newmark, three kinds of space–time adaptivity strategies based on hierarchical a-posteriori error estimations are developed successively, that is, Unidirectional Multi-level Space-Time Adaptive GIFT/Newmark (UM-STAGN), Energy-based Space-Time Adaptive GIFT/Newmark (E-STAGN), and Goal-oriented Space-Time Adaptive GIFT/ Newmark (G-STAGN) methods respectively. The main concept of UM-STAGN approach is to get rid of the elements where the error estimators reach the prescribed accuracy at each adaptation step, and then assemble the rest elements as the new subsystems to be solved at the next stage. By reducing the scale of the computational domains gradually, this method achieves efficiency to some degree, though, since it fails to transfer the error information across subsystems, the error of Quantity of Interest (QoI) cannot arrive at an expected precision. For this reason, we introduce E-STAGN methodology, wherein the error indicator of each element is reassessed at every adaptive cycle through solving the whole system again. In this case, the QoI error is able to be convergent to an acceptable accuracy. Nevertheless, as the error is accumulated with time in the time-domain problem, E-STAGN method based on the energy-norm error estimation is unable to expose the source of the error so that it fails to offer an efficient refinement. This is the motivation to establish the G-STAGN technique, where the error estimation drives from our proposed first-order Dual Weight Residual (DWR) method. G-STAGN strategy can detect the origin of QoI error, and hence it leads to a remarkably economical refinement. Numerical examples are carried out in both single-patch and multi-patch structures. It is demonstrated that both UM-STAGN and E-STAGN methods can catch the propagation of stress wave for the primal problem, while G-STAGN technique is capable to track the travel path of dual stress. Therefore, the G-STAGN strategy achieves an optimal convergent rate, compared to that obtained by UM-STAGN, E-STAGN and uniform space–time h-refinement approaches.

An adaptive variational multiscale element free Galerkin method based on the residual-based a posteriori error estimators for convection–diffusion–reaction problems

As we know, the variational multiscale element free Galerkin (VMEFG) method may still suffer from non-physical oscillations near the boundary or interior layers when solving the convection–diffusion–reaction problems with strong convection-dominated. To overcome this shortcoming, we consider incorporating an adaptive algorithm based on the residual error estimations into the VMEFG, which formed the adaptive VMEFG (AVMEFG) method. With respect to the adaptive technique adopted, two residual-based a posteriori error estimators in the H1-semi norm and energy norm are respectively used to locate and remark the high-gradient numerical solution regions. Several stationary convection–diffusion–reaction problems are solved to verify the effectiveness of the proposed method. Among them, the first example makes a comparison between the proposed method and the adaptive element free Galerkin method, while the other remaining examples compare two different residual-based a posteriori error estimators for the proposed method. Numerical examples illustrate that the proposed method is effective and efficient in solving the convection-dominated problem involving various layers. Moreover, the AVMEFG methods with the two residual-based a posteriori error estimators are generally equivalent except in the case that the solution has an exponential boundary layer, where the energy norm error estimator can achieve significantly better results than the other one.

Error estimate of a decoupled numerical scheme for the Cahn–Hilliard–Stokes–Darcy system

Abstract We analyze a fully discrete finite element numerical scheme for the Cahn–Hilliard–Stokes–Darcy system that models two-phase flows in coupled free flow and porous media. To avoid a well-known difficulty associated with the coupling between the Cahn–Hilliard equation and the fluid motion, we make use of the operator-splitting in the numerical scheme, so that these two solvers are decoupled, which in turn would greatly improve the computational efficiency. The unique solvability and the energy stability have been proved in Chen et al. (2017, Uniquely solvable and energy stable decoupled numerical schemes for the Cahn–Hilliard–Stokes–Darcy system for two-phase flows in karstic geometry. Numer. Math., 137, 229–255). In this work, we carry out a detailed convergence analysis and error estimate for the fully discrete finite element scheme, so that the optimal rate convergence order is established in the energy norm, i.e., in the $\ell ^{\infty } (0, T; H^1) \cap \ell ^2 (0, T; H^2)$ norm for the phase variables, as well as in the $\ell ^{\infty } (0, T; H^1) \cap \ell ^2 (0, T; H^2)$ norm for the velocity variable. Such an energy norm error estimate leads to a cancelation of a nonlinear error term associated with the convection part, which turns out to be a key step to pass through the analysis. In addition, a discrete $\ell ^2 (0;T; H^3)$ bound of the numerical solution for the phase variables plays an important role in the error estimate, which is accomplished via a discrete version of Gagliardo–Nirenberg inequality in the finite element setting.

Improvement in 3D topology optimization with h‐adaptive refinement using the Cartesian grid Finite Element Method

AbstractThe growing number of scientific publications on topology optimization (TO) shows the great interest that this technique has generated in recent years. Among the different methodologies for TO, this article focuses on the well‐known solid isotropic material penalization (SIMP) method, broadly used because of its simple formulation and efficiency. Even so, the SIMP method has certain drawbacks, namely: lack of precision in definition of the edges of the optimized geometry and final results strongly influenced by the discretization used for the finite element (FE) analyses. In this article, we propose a combination of techniques to limit the effect of these drawbacks and, thus, to improve the behavior of TO. All these techniques are based on the use of the Cartesian grid finite element method (cgFEM), an immersed boundary method whose Cartesian grid structure and hierarchical data structure makes it specially appropriate for TO. All the proposed techniques are framed under the concept of mesh refinement. First, we propose the use of two meshes, the FE analysis mesh, and a finer mesh for integration and evaluation of sensitivities, to improve the resolution of the final solution at a marginal computational cost. Then we propose two h‐adaptive mesh refinement strategies. The first one will tend to refine the elements having intermediate density values and will have the effect of sharpening the definition of the edges of the optimized geometry. We will clearly show that if the accuracy of the FE analyses is not taken into account, stress constrained TO will generate solutions that, once manufactured, will not satisfy the constraints. Hence, we also propose an h‐refinement strategy based on the estimation of the discretization error in energy norm.

A posteriori error estimation for numerical model reduction in computational homogenization of porous media

SummaryNumerical model reduction is adopted for solving the microscale problem that arizes from computational homogenization of a model problem of porous media with displacement and pressure as unknown fields. A reduced basis is obtained for the pressure field using (i) spectral decomposition (SD) and (ii) proper orthogonal decomposition (POD). This strategy has been used in previous work—the main contribution of this article is the extension with an a posteriori estimator for assessing the error in (i) energy norm and in (ii) a given quantity of interest. The error estimator builds on previous work by the authors; the novelty presented in this article is the generalization of the estimator to a coupled problem, and, more importantly, to accommodate the estimator for a POD basis rather than the SD basis. Guaranteed, fully computable and low‐cost bounds are derived and the performance of the error estimates is demonstrated via numerical results.

Superconvergence analysis of a nonconforming finite element method for monotone semilinear elliptic optimal control problems

AbstractA nonconforming finite element method (FEM) is proposed for optimal control problems (OCPs) governed by monotone semilinear elliptic equations. The state and adjoint state are approximated by the nonconforming elements, and the control is approximated by the orthogonal projection of the adjoint state, respectively. Some global supercloseness and superconvergence estimates are achieved by making full use of the distinguish characters of this element, such as the interpolation equals to its Ritz projection, and the consistency error is 1 − ε ( is small enough) order higher than its interpolation error in the broken energy norm when the exact solution belongs to H3 − ε(Ω). Finally, some numerical results are presented to verify the theoretical analysis.

Guaranteed a posteriori error bounds for low-rank tensor approximate solutions

Abstract We propose a guaranteed and fully computable upper bound on the energy norm of the error in low-rank tensor train (TT) approximate solutions of (possibly) high-dimensional reaction–diffusion problems. The error bound is obtained from Euler–Lagrange equations for a complementary flux reconstruction problem, which are solved in the low-rank TT representation using the block alternating linear scheme. This bound is guaranteed to be above the energy norm of the total error, including the discretization error, the tensor approximation error and the error in the solver of linear algebraic equations, although quadrature errors, in general, can pollute its evaluation. Numerical examples with the Poisson equation and the Schrödinger equation with the Henon–Heiles potential in up to 40 dimensions are presented to illustrate the efficiency of this approach.

The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering

Abstract In this study, for the first time, we discuss the posteriori error estimates and adaptive algorithm for the non-self-adjoint Steklov eigenvalue problem in inverse scattering. The differential operator corresponding to this problem is non-self-adjoint and the associated weak formulation is not H 1-elliptic. Based on the study of Armentano et al. [Appl. Numer. Math. 58 (2008), 593–601], we first introduce error indicators for primal eigenfunction, dual eigenfunction, and eigenvalue. Second, we use Gårding’s inequality and duality technique to give the upper and lower bounds for energy norm of error of finite element eigenfunction, which shows that our indicators are reliable and efficient. Finally, we present numerical results to validate our theoretical analysis.

Discontinuous Galerkin methods for fractional elliptic problems

The aim of this paper is to provide a mathematical framework for studying different versions of discontinuous Galerkin (DG) approaches for solving 2D Riemann–Liouville fractional elliptic problems on a finite domain. The boundedness and stability analysis of the primal bilinear form are provided. A priori error estimate under energy norm and optimal error estimate under $$L^{2}$$ norm are obtained for DG methods of the different formulations. Finally, the performed numerical examples confirm the optimal convergence order of the different formulations.

Global superconvergence analysis of a nonconforming FEM for Neumann boundary OCPs with elliptic equations

In this paper, a nonconforming finite element method (FEM) is proposed for the Neumann type boundary optimal control problems (OCPs) governed by elliptic equations. The state and adjoint state are approximated by the nonconforming elements, and the control is approximated by the orthogonal projection through the adjoint state. Some superclose behaviors are derived by full use of the distinguish characters of this element, such as the interpolation operator equals to the Ritz projection, and the consistency error is higher than its interpolation error in the broken energy norm. After that, the global superconvergence results are obtained by employing the so-called post-interpolation technique. Finally, some numerical results are provided to verify the theoretical analysis.

Selective enrichment of moment fitting and application to cut finite elements and cells

For many computational methods, the important task of integrating discontinuous functions can turn out to be a bottleneck. In extended finite element methods as well as in fictitious domain or immersed boundary schemes or cut finite elements, the discretization of the problem of interest involves the computation of integrals with discontinuous functions. To this end, we propose an efficient moment fitting scheme applicable to high-order methods, accounting for discontinuities on curved interfaces. The integration points are pre-selected as the standard Gauss–Legendre abscissae, and the basis functions of the moment fitting are defined in such a way that solving an equation system can be circumvented, thus reducing the overhead of the proposed approach significantly. An example related to a mechanical problem of a 1D bar exhibiting a material interface serves to demonstrate the possibility of adjusting the enrichment of the basis function of the moment fitting with the trial and test function space of the Galerkin method. Due to the multiplicative or tensor product nature of the set of basis functions to be integrated, a selective approach is proposed in order not to increase the number of integration points. In this way, high precision can be achieved for a p-extension of the spatial discretization, yielding an exponential convergence of the error in energy norm without the necessity of refining the mesh. The computation of the required quadrature rule for the discontinuous integrand is carried out in an offline stage. The related overhead is relatively small and amortized during the evaluation of costly integrals such as the computation of high-order stiffness matrices. The approach should lead to a huge time saving, especially in the case of nonlinear problems where the quadrature rule can be reused many times. Two-dimensional and three-dimensional problems show that the proposed moment fitting can also be applied for the integration of high-order functions including discontinuities across curved boundaries.