L2 Norm Of Error Research Articles

XFEM analysis for effectively modeling the singularity of the capacitor edge

In this paper, the edge effect of a parallel plate capacitor is numerically analyzed using the extended finite element method, which exhibits higher accuracy than the standard finite element method. The enrichment function, which plays an important role in this method, is numerically computed on the virtual plane in a simple manner using conformal mapping and complex number techniques. The complex velocity potential is used to derive the gradient of the enrichment function. In the implementation process, the inverse of the conformal mapping is derived numerically via triangular patches and the normalized coordinates within them because analytical derivation is challenging. Moreover, the coordinates are converged using the complex Newton–Raphson iterative method. The effectiveness of the proposed method is verified through numerical experiments. The L2 error norm of solution is used to evaluate the accuracy quantitatively. The proposed method exhibits smaller error norms in comparison with the standard finite element method.

One-dimensional heat and advection-diffusion equation based on improvised cubic B-spline collocation, finite element method and Crank-Nicolson technique

In this study, we numerically investigate the solution of one-dimensional heat and advection-diffusion equation by employing improvised quartic order cubic B-spline using Crank-Nicolson method and finite element method (FEM). The numerical technique discretizes the spatial derivatives variables using Crank-Nicolson method and time derivatives using finite element method. It is demonstrated that the method is unconditionally stable with Von-Neumann process and accurate to convergence order Oh4+∆t2. The numerical results are simulated and error bounds are calculated using finite difference scheme on a piecewise uniform mesh. Finally, to support our claim, five test problems are considered and the experimental results also compared to existing methods using MATLAB as well as MATHEMATICA software tools. The error norms L2,L∞,root mean square (RMS) error with computational time and order of convergence are investigated for each example. The method is computationally efficient with less memory storage. The accuracy and efficiency of present scheme is measured in terms of absolute errors and also compared with analytical and published results in literature.

L2 error estimates for a family of cubic finite volume methods on triangular meshes

This paper considers the error estimates of a family of Lagrange cubic finite volume methods for solving two dimensional elliptic boundary value problems over triangular meshes. By introducing a novel map of finite volume method, we propose a family of cubic finite volume method schemes with four independent map coefficients. It is detailedly proved that the proposed finite volume method schemes is exactly equivalent to the existing traditional finite volume method schemes of the same boundary value problems. A geometric constraint requirement on the triangular meshes of primary partitions is imposed to guarantee the uniform ellipticity of discrete elliptic bilinear forms resulted from the proposed schemes. The geometric requirements of the primary partitions could be greatly attenuated when the map coefficients are appropriately designed. Once the uniform ellipticity of the discrete elliptic bilinear forms is established, the proposed schemes have the unique solvability, and the optimal H1-norm error estimates without any restriction on dual elements of dual partitions of the finite volume methods, which is consistent with the results of literatures. With the help of the introduced map, the proposed schemes have the optimal L2-norm error estimates with a mesh restriction for the dual elements of their dual partitions significantly weaker than the existing theoretical results, in which the Aubin-Nitsche duality argument and the orthogonal conditions based on the dual partitions are employed. Finally, we validate our theoretical results by numerical experiments.

The Convergence of Symmetric Discretization Models for Nonlinear Schrödinger Equation in Dark Solitons’ Motion

The Schrödinger equation is one of the most basic equations in quantum mechanics. In this paper, we study the convergence of symmetric discretization models for the nonlinear Schrödinger equation in dark solitons’ motion and verify the theoretical results through numerical experiments. Via the second-order symmetric difference, we can obtain two popular space-symmetric discretization models of the nonlinear Schrödinger equation in dark solitons’ motion: the direct-discrete model and the Ablowitz–Ladik model. Furthermore, applying the midpoint scheme with symmetry to the space discretization models, we obtain two time–space discretization models: the Crank–Nicolson method and the new difference method. Secondly, we demonstrate that the solutions of the two space-symmetric discretization models converge to the solution of the nonlinear Schrödinger equation. Additionally, we prove that the convergence order of the two time–space discretization models is O(h2+τ2) in discrete L2-norm error estimates. Finally, we present some numerical experiments to verify the theoretical results and show that our numerical experiments agree well with the proven theoretical results.

Nonsymmetric interior penalty Galerkin method for nonlinear time-fractional integro-partial differential equations

This article discusses an efficient numerical method for solving nonlinear time-fractional integro-partial differential initial–boundary-value problems. To tackle the non-linearity of the problem, we first use the Newton linearization process. Then we apply the non-symmetric interior penalty Galerkin method for the spatial variable, and obtain a semi-discrete problem in the time variable. Finally, to obtain the fully-discrete scheme, we use both L1-scheme and L2-scheme for time-fractional derivative, and trapezoidal rule for integral term in the semi-discrete problem. We derive L2-norm stability and error estimates. The validity of the error analysis is supported by numerical experiments.

Data-driven multi-valley dark solitons of multi-component Manakov Model using Physics-Informed Neural Networks

In this paper, we employ a Deep Learning technique, namely Physics-Informed Neural Network for solving multi-component Manakov models. In particular, we consider three and four-coupled nonlinear Schrödinger equations. We show that in the former case, one of the components and two others can carry, respectively, a double valley dark soliton and asymmetric bright solitons. In the four-component Manakov model, all four components are predicted by PINN with high accuracy in which the first component is shown to be a triple-valley dark soliton. We compare the results with the exact solutions and bring out the ability of Deep Learning in solving coupled systems of nonlinear partial differential equations with high accuracy. The performance of the PINN in approximating the solutions is estimated by taking the squared difference between the exact and predicted squared magnitude of the solutions. We have also computed the L2-norm errors for real, imaginary and absolute-value components of the solutions to emphasize the accuracy of PINN.

Unconditional optimal error estimates of a linearized mass- and energy- conservation FEM for a coupled nonlinear Schrödinger equations

In this paper, we study a linearized finite element method for a coupled nonlinear Schrödinger system (CNLS). We apply a modified leap-frog scheme for time discretization and a Galerkin finite element method for spatial discretization. We prove that this method is conservative for mass and energy in a modified form. By an error splitting technique, we split the error into two parts, the temporal error and the spatial error. First we prove the uniform boundedness in H2-norm for the solutions of the time-discrete system, and obtain the temporal error estimates. Second, we prove the pointwise uniform boundedness of the finite element solution. Then we get the optimal L2-norm error estimates without any time-step restriction. Finally, in order to confirm our theoretical analysis, we provide numerical examples to validate the convergence-order, unconditional stability and conservation for mass and energy.

Simulations of the one and two dimensional nonlinear evolutionary partial differential equations: A numerical study

In this work a hybrid scheme is proposed for the numerical study of various evolutionary partial differential equations (EPDEs). In proposed strategy, temporal derivatives are estimated via first and second order finite differences while the solution and spatial derivatives are approximated by Lucas and Fibonacci polynomials. Further, the collocation approach is applied which convert the EPDEs, to the system of coupled linear equations which are simple to solve. For non-linear problems, Quasilinearization is used to tackle nonlinearity. The scheme is implemented to solve different EPDEs which include, one dimensional non-linear Boussinesq, Hunter–Saxton, wave-like and two dimensional linear and non-linear wave like equations. Accuracy of the scheme is portrayed by computing L∞,L2 error norms and the relative error. Moreover, the calculated outcomes are compared with previously existing results in literature. Simulation demonstrates that the scheme works well for the mentioned EPDEs.

Multi-scale Generative Adversarial Deblurring Network with Gradient Guidance

<>With regards to the lack of crisp edges and a poor recovery of high frequency information such as details in deblurred motion pictures, this research proposes a multi-scale adversarial deblurring network with gradient guidance (MADN). The algorithm uses the classical generative adversarial network (GAN) framework, consisting of a generator and a discriminator. The generator includes a multi-scale convolutional network and a gradient feature extraction network. The multi-scale convolutional network extracts image features at different scales with a nested connection residual codec structure to improve the image edge structure recovery and to increase the perceptual field. This gradient network incorporates with intermediate scale features to extract the gradient features of blurred images to obtain their high frequency information. The generator combines the gradient and multiscale features to recover the remaining high-frequency information in a deblurred image. The loss function of MADN is formed in this research combining adversarial loss, pixel L2-norm loss and mean absolute error. Compared to those experimental results obtained from current deblurring algorithms, our experimental results indicate visually clearer images retaining more information such as edges and details. This MADN algorithm enhances the peak signal-to-noise ratio by an average of 3.32dB and the structural similarity by an average of 0.053.<>

Variable-time-step BDF2 nonconforming VEM for coupled Ginzburg-Landau equations

In this paper, we solve the coupled Ginzburg-Landau equations by using a linearized variable-time-step second order backward differentiation formula in time combining with a nonconforming virtual element method in space. Based on the techniques of the discrete orthogonal convolution kernels and the discrete complementary convolution kernels, the unconditional optimal L2-norm error estimate of the fully discrete scheme was presented by using the error splitting technique under the mild restriction on the ratio of adjacent time-steps ratios. Finally, numerical experiments illustrate the correctness of our theoretical analysis.

Verification of the generalized reduced-order particle-in-cell scheme in a radial–azimuthal E × B plasma configuration

In this article, we present an in-depth verification of the generalized electrostatic reduced-order particle-in-cell (PIC) scheme in a cross electric and magnetic field configuration representative of a radial–azimuthal section of a Hall thruster. The setup of the simulations follows a well-established benchmark case. The main purpose of this effort is to demonstrate that our novel PIC scheme can reliably resolve the complex two-dimensional dynamics and interactions of the plasma instabilities in the radial–azimuthal coordinates of a Hall thruster at a fraction of the computational cost compared to full-2D PIC codes. To this end, we first present the benchmarking of our newly developed full-2D PIC code. Next, we provide an overview of the reduced-order PIC scheme and the resulting “quasi-2D” code, specifying that the degree of order reduction in the quasi-2D PIC is defined in terms of the number of “regions” along the simulation’s directions used to divide the computational domain. We compare the predictions of the quasi-2D simulation in various approximation degrees of the 2D problem against our full-2D simulation results. We show that, with an increase in the number of regions in the Q2D simulations, the quasi-2D results converge to the 2D ones. Nonetheless, we also highlight that a quasi-2D simulation that provides a factor of 5 reduction in the computational cost resolves the underlying physical processes in an almost indistinguishable manner with respect to the full-2D simulation and incurs an L2-norm error of only about 2% in the ion number density and below 1% in the electron temperature.

An accuracy analysis framework for meshfree collocation methods with particular emphasis on boundary effects

A theoretical framework for the accuracy analysis of meshfree collocation methods is presented with a particular focus on the effects of boundary conditions. In the proposed framework, the error of a meshfree collocation formulation is decomposed into three parts, namely, the collocation errors associated with the interior nodes, the Dirichlet boundary nodes and the Neumann boundary nodes, respectively. The errors arising from different parts of the computational domain can be conveniently obtained by performing a local truncation error analysis on the discrete meshfree collocation equations corresponding to the generic interior and boundary nodes. It turns out that the accuracy of a meshfree collocation formulation is controlled by the lowest accuracy order resulting from the collocation errors at both interior and boundary nodes, as well as the meshfree interpolation errors. Along this path, it is shown that accuracy of the standard node-based meshfree collocation method is governed by the collocation error related to interior nodes for both L2 and H1 error norms. Nonetheless, for different boundary conditions, the accuracy of the subdomain meshfree collocation method may rely on either the collocation error associated with interior nodes, the collocation errors corresponding to Dirichlet and Neumann boundary nodes, or the interpolation errors. It is further proved that the optimal convergence rates of Galerkin meshfree formulation can be achieved by the subdomain meshfree collocation method if the problems are purely subjected to Dirichlet boundary conditions. The proposed theoretical results are well confirmed by numerical examples.

Prediction of surface morphology and reflection spectrum of laser-induced periodic surface structures using deep learning

Laser-induced periodic surface structures have been extensively explored as an enabling tool for fabricating various optical surfaces because the resulting surface ripples can effectively modify surface reflectance. Here, we propose a deep-learning-based model for predicting high-quality surface scanning electron microscopy (SEM) images with detailed surface morphology corresponding to unexplored process conditions. In addition, the reflectance value at 550 nm and reflection spectra from the generated (or original) SEM images were predicted. To obtain training data, stainless steel 304 specimens were processed with femtosecond laser pulses on a large process window consisting of 32 process conditions to obtain SEM images with various surface morphologies and corresponding reflection spectra. The image prediction model is based on a conditional generative adversarial network, which generates a surface SEM image from the laser fluence and scanning speed values. The reflectance prediction model was developed based on ResNet152 and Long-Short Term Memory network. The average accuracies for the pattern period and ripple width were 98.2 and 94.6 %, respectively, and the L2 norm error for the reflection spectra was less than 4 %. It was demonstrated that the reflection spectra can be accurately predicted using only surface images, which can also be accurately generated from process parameters.

Two energy stable variable-step L1 schemes for the time-fractional MBE model without slope selection

A variable-step L1 scheme is proposed for the time-fractional molecular beam epitaxy model without slope selection. By taking advantage of the convex splitting of nonlinear bulk, a sharp L2 norm error estimate is established under a convergence–solvability–stability (CSS) consistent time-step constraint, that is, the maximum step-size limit required for convergence is of the same order to that for solvability and stability (in certain norms) as the small interface parameter ϵ→0+. By using the discrete gradient structure of L1 formula, we build up a discrete variational energy dissipation law, which is asymptotically compatible with the classical energy dissipation law, as the fractional order α→1−. The stability and convergence of the stabilized convex splitting scheme are also investigated. The resulting scheme is shown to unconditionally preserve the variational energy dissipation law by adding a (2−α)-order stabilization term like τn1−α(ϕn−ϕn−1). Numerical experiments are presented to support our theoretical results.

NUMERICAL SIMULATION OF GENERALIZED OSKOLKOV EQUATION VIA THE SEPTIC B-SPLINE COLLOCATION METHOD

In this paper, one of the nonlinear evolution equation (NLEE) namely generalised Oskolkov equation which defines the dynamics of an incompressible visco-elastic Kelvin-Voigt fluid is investigated. We discuss numerical
 solutions of the equation for two test problems including single solitary wave and Gaussian initial condition, applying the collocation finite element method. The algorithm, based upon Crank Nicolson approach in time, is unconditionally stable. To demonstrate the proficiency and accuracy of the numerical algorithm, error norms L2, L∞ and invariant I are calculated and the obtained results are indicated both in tabular and graphical form. The obtained numerical results provide the method is more suitable and systematically handle the solution procedures of nonlinear equations arising in mathematical physics.

Compatible L2 norm convergence of variable-step L1 scheme for the time-fractional MBE model with slope selection

The convergence of variable-step L1 scheme is studied for the time-fractional molecular beam epitaxy (MBE) model with slope selection. A novel asymptotically compatible L2 norm error estimate of the variable-step L1 scheme is established under a convergence-solvability-stability (CSS)-consistent time-step constraint. The CSS-consistent condition means that the maximum step-size limit required for convergence is of the same order to that for solvability and stability (in certain norms) as the small interface parameter ϵ→0+. To the best of our knowledge, it is the first time to establish such error estimate for nonlinear subdiffusion problems. The asymptotically compatible convergence means that the error estimate is compatible with that of backward Euler scheme for the classical MBE model as the fractional order α→1−. Just as the backward Euler scheme can maintain the physical properties of the MBE equation, the variable-step L1 scheme can also preserve the corresponding properties of the time-fractional MBE model, including the volume conservation, variational energy dissipation law and L2 norm boundedness. Numerical experiments are presented to support our theoretical results.

A collocation algorithm based on septic B-splines and bifurcation of traveling waves for Sawada–Kotera equation

In present work, a mathematical model standing for the numerical solutions of the nonlinear Sawada–Kotera (S–K) equation has been examined. For this purpose, we have used collocation finite element method for the model problem. The solution for each unknown is written as a linear combination of time parameters and combination of the septic B-splines. Next, applying Von-Neumann theory, we demonstrate that the scheme is marginally stable. Error norms L2 and L∞ are computed for single soliton solutions to display the practicality and robustness the suggested scheme. The obtained numerical results have been illustrated with tables. Furthermore, to show the numerical behavior of the single soliton, some of the simulations are depicted in 2D and 3D. Present results show that the method procures high accurate solutions so present scheme will be useful for the other nonlinear scientific problems. After that, bifurcation behavior of traveling waves for the S-K equation has been examined depending on the velocity (v) of the traveling wave. It has been seen that velocity (v) of the traveling wave plays a key role for the existence of different kind of quasiperiodic motions of the nonlinear S-K equation. Coexisting quasiperiodic motions for traveling wave solutions of the equation have been presented.

Global superconvergence analysis of nonconforming finite element method for time fractional reaction-diffusion problem with anisotropic data

In this paper, a class of two-dimensional (2-D) time fractional reaction-diffusion equation is considered. The solution usually exhibits singularity at the initial moment and anisotropic behavior in the spatial direction. In response to these problems, we provide an effective numerical framework for analyzing the L2-norm error, H1-norm superclose property and H1-norm global superconvergence result. This framework combines the high-precision L2-1σ scheme on non-uniform time grids and the anisotropic nonconforming quasi-Wilson finite element method (FEM) in space. Some numerical experiments are presented to illustrate our theoretical findings.

On the role of bond-associated stabilization and discretization on deformation and fracture in non-ordinary state-based peridynamics

As it is well known, non-ordinary state-based peridynamic (NOSB PD) models suffer from inherent instability issues, which can severely affect deformation and fracture predictions. These issues have been identified and tackled through the years using stabilization methods that work by introducing additional terms in the formulation or by disturbing the radial symmetry of the horizon, leading to stabilized solutions. A popular and effective method is the bond-associated correspondence formulation. In this work, we make an in-depth investigation on the role of bond-associated stabilization and discretization on the deformation and fracture behaviour in NOSB PD by considering three different comprehensive and detailed numerical studies. Firstly, the convergence of the NOSB PD linear elastic formulation is investigated under both the L2 and H1 error norms. The role of the influence function is also considered and analysed. Different combinations of bond-associated horizon and point-associated horizon values are used to determine the role of these parameters in obtaining accurate solutions. Secondly, we consider a static analysis example where we compare PD with the finite element method (FEM), in order to verify the accuracy of the PD models for stress analysis. Lastly, a quasi-static crack propagation analysis benchmark example is presented to assess the accuracy of fracture predictions and the effect of varying the discretization parameters is studied for both load–displacement curves and crack paths.

A second-order scheme with nonuniform time grids for Caputo–Hadamard fractional sub-diffusion equations

In this paper, a second-order scheme with nonuniform time meshes for Caputo–Hadamard fractional sub-diffusion equations with initial singularity is investigated. Firstly, a Taylor-like formula with integral remainder is proposed, which is crucial to studying the discrete convolution kernels and error estimate. Secondly, we come up with an error convolution structure (ECS) analysis for Llog,2−1σ interpolation approximation to the Caputo–Hadamard fractional derivative. The core result in this paper is an ECS bound and a global consistency analysis established at an offset point. By virtue of this result, we obtain a sharp L2-norm error estimate of a second-order Crank–Nicolson-like scheme for Caputo–Hadamard fractional differential equations. Ultimately, an example is presented to show the sharpness of our analysis.