L2 Norm Of Error Research Articles

Modified approximation for the KdV–Burgers equation

In this paper, a new finite difference called the Restrictive Taylor approximation (RTA) is implemented to find numerical solution of KdV-Burgers, this method is a new explicit method. The accuracy of the method is assessed in terms of the absolute error which is very close to zero and the error norms L2, L∞.

Numerical Method Using Cubic B-Spline for a Strongly Coupled Reaction-Diffusion System

In this paper, a numerical method for the solution of a strongly coupled reaction-diffusion system, with suitable initial and Neumann boundary conditions, by using cubic B-spline collocation scheme on a uniform grid is presented. The scheme is based on the usual finite difference scheme to discretize the time derivative while cubic B-spline is used as an interpolation function in the space dimension. The scheme is shown to be unconditionally stable using the von Neumann method. The accuracy of the proposed scheme is demonstrated by applying it on a test problem. The performance of this scheme is shown by computing and error norms for different time levels. The numerical results are found to be in good agreement with known exact solutions.

Numerical study of staggered scheme for viscous Saint-Venant equations

This paper describes a numerical scheme for approximate the viscous Saint-Venant equations. This scheme is called staggered grid scheme which is a robust, simple and strightforward scheme for viscous SaintVenant equations. Some numerical simulations have been elaborated to validate the accuracy of the scheme, such as the calculation of the convergence rate L1-norm error of the scheme, the comparison of viscous and classical Saint-Venant equations, comparison of staggered scheme with kinetic scheme and direct Navier-Stokes approach in dam-break problem and simulation of dry-wet problem. The results show good accuracy and in a good agreement with another scheme in another paper. Mathematics Subject Classification: 35C20, 76B07, 76B25, 76M20

Optimal multi-degree reduction of Bézier curves with geometric constraints

In this paper we present a novel algorithm for the multi-degree reduction of Bézier curves with geometric constraints. Based on the given constraints, we construct an objective function which is abstracted from the approximation error in L2-norm. Two types of geometric constraints are tackled. With the constraints of G2-continuity at one endpoint and G1-continuity (or Cr-continuity) at the other endpoint, we derive the optimal degree-reduced curves in explicit form. With the constraints of G2-continuity at two endpoints, the problem of degree reduction is equivalent to minimizing a bivariate polynomial function of degree 4. Compared with the traditional methods, we derive the optimal degree-reduced curves more effectively. Finally, evaluation results demonstrate the effectiveness of our method.

A class of finite volume schemes of arbitrary order on nonuniform meshes

AbstractIn this article, we generalize the bi‐ finite volume schemes developed in Zhang and Zou, J Sci Comput (in press) for elliptic equations with high smooth solutions to elliptic equations with singular solutions. By designing a special nonuniform rectangular meshes, we construct a class of finite volume schemes of arbitrary order k. Our theoretic analysis shows that if the solution has weak singularity of type , where r is the distance from some target point to some fixed singular point, the H1 norm of our finite volume schemes' discretization error converges with optimal order , while the L2 norm error converges with order . Here, N2 is the cardinality of the underlying mesh. Superconvergence property of the scheme is also discussed. Our theoretic findings have been verified by numerical experiments. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1614–1632, 2014

Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems

We study the spectral approximation properties of finite element and NURBS spaces from a global perspective. We focus on eigenfunction approximations and discover that the L2-norm errors for finite element eigenfunctions exhibit pronounced “spikes” about the transition points between branches of the eigenvalue spectrum. This pathology is absent in NURBS approximations. By way of the Pythagorean eigenvalue error theorem, we determine that the squares of the energy-norm errors of the eigenfunctions are the sums of the eigenvalue errors and the squares of the L2-norm eigenfunction errors. The spurious behavior of the higher eigenvalues for standard finite elements is well-known and therefore inherited by the energy-norm errors along with the spikes in the L2-norm of the eigenfunction errors. The eigenvalue pathology is absent for NURBS. The implications of these results to the corresponding elliptic boundary-value problem and parabolic and hyperbolic initial-value problems are discussed.

Linear reconstruction measure steered nearest neighbor classification framework

The linear reconstruction measure (LRM), which determines the nearest neighbors of the query sample in all known training samples by sorting the minimum L2-norm error linear reconstruction coefficients, is introduced in this paper. The intuitive interpretation and mathematical proofs are presented to reveal the efficient working mechanism of LRM. Through analyzing the physical meaning of coefficients and regularization items, we find that LRM provides more useful information and advantages than the conventional similarity measure model which calculates the distance between two entities (i.e. conventional point-to-point, C-PtP). Inspired by the advantages of LRM, the linear reconstruction measure steered nearest neighbor classification framework (LRM-NNCF) is designed with eight classifiers according to different decision rules and models of LRM. Evaluation on several face databases and the experimental results demonstrate that these proposed classifiers can achieve greater performance than the C-PtP based 1-NNs and competitive recognition accuracy and robustness compared with the state-of-the-art classifiers.

An implicit [formula omitted] multi patch B-spline interpolation for Kirchhoff–Love space rod

The paper presents a novel formulation for the isogeometric analysis of assemblies of Kirchhoff–Love space rod elements, introducing a multi-patch implicit G1 formulation, so that an automatic non-singular stiffness operator is obtained without the need of adding continuity conditions. The goal is achieved using a polar decomposition of the deformation of the first and last segments of the control polygon, that allows to introduce directly the end rotations as degrees of freedom. Both parametric and geometric continuity can be obtained in this way. We use Bezier and B-spline interpolations and we show that they are able to attain very good accuracy for developing a 3D exact curve element with geometric torsion (pre-twisted rod). In the paper the performance of the multi-patch elements is examined comparing the rates of convergence of the L2 error norm for the multi-patch and single-patch formulations. It is shown that the rate of convergence remains the same, although in certain cases the accuracy is lower for the multi-patch solutions.

An approximation of semiconductor device by mixed finite element method and characteristics-mixed finite element method

The mathematical model of a semiconductor device is described by a system of three quasi-linear partial differential equations for initial boundary value problem. The electric potential is governed by an elliptic equation. The electron and hole concentrations are governed by two equations of convection-dominated diffusion type. Standard mixed finite element is used for the electric potential equation. A characteristics-mixed finite element method is presented for the concentration equations. This scheme conserves mass locally. In order to derive the optimal L2-norm error estimates, a post-processing step is included in the approximation to the scalar concentration. Numerical experiment is presented finally to validate the theoretical analysis.

A Numerical Solution of the MEW Equation Using Sextic B-Splines

In this article, a numerical solution of the modified equal width wave (MEW) equation, based on subdomain method using sextic B-spline is used to simu- late the motion of single solitary wave and interaction of two solitary waves. The three invariants of the motion are calculated to determine the conservation properties of the system. L2 and L1 error norms are used to measure differences between the analyti- cal and numerical solutions. The obtained results are compared with some published numerical solutions. A linear stability analysis of the scheme is also investigated.

Strong form Meshless Implementation of Taylor Series Method

A new meshless approach is investigated by using Taylor series expansion and technique of differential transform method, which is called Meshless Implementation of Taylor Series Method (MITSM). In particular, Strong form Meshless Implementation of Taylor Series Method (SMITSM) is studied in this paper. Then, the basis functions are used to solve a 1D second-order ordinary differential equation and 2D Laplace equation by using the SMITSM. Comparisons are made with the analytical solutions and results of Symmetric Smoothed Particle Hydrodynamics (SSPH) method. We also compared the effectiveness of the SMITSM and SSPH method by considering various particle distributions, nonhomogeneous terms and number of terms in the basis functions. It is observed that the MITSM has the conventional convergence properties and, at the expense of CPU time, yields smaller L2 error norms than the SSPH method, especially in the existence of nonsmooth nonhomogeneous problems.

An approximation of semiconductor device of heat conduction by mixed finite element method and characteristics-mixed finite element method

The mathematical model for semiconductor devices of heat conduction is numerically discretized. The physical variables are the electrostatic potential, the electron and hole concentrations, and the temperature. Standard mixed finite element is used for the elliptic electric potential equation. A characteristics-mixed finite element method is presented for the two convection-dominated concentration equations. Standard finite element is used for the temperature equation of parabolic type. This scheme conserves mass locally for the concentrations. In order to derive the optimal L2-norm error estimates, a post-processing step is included in the approximation to the scalar concentrations. Numerical experiment is presented finally to validate the theoretical analysis.

Crank-Nicolson split least-squares procedure for nonlocal reactive flow in porous media

In this paper, we introduce a Crank-Nicolson split least-squares Galerkin finite element procedure for parabolic integro-differential equations, arising in the modeling of nonlocal reactive flows in porous media. By selecting the least-squares functional properly, the procedure can be split into two independent sub-procedures, one of which is for the primitive unknown and the other is for the flux. By carefully choosing projections, we get optimal order H1(Ω) and L2(Ω) norm error estimates for u and sub-optimal (L2(Ω))d norm error estimate for σ with second-order accuracy in time increment. The numerical examples are given to testify the efficiency of the introduced scheme.

Error estimates for the ultra weak variational formulation in linear elasticity

We prove error estimates for the ultra weak variational formulation (UWVF) in 3D linear elasticity. We show that the UWVF of Navier’s equation can be derived as an upwind discontinuous Galerkin method. Using this observation, error estimates are investigated applying techniques from the theory of discontinuous Galerkin methods. In particular, we derive a basic error estimate for the UWVF in a discontinuous Galerkin type norm and then an error estimate in the L2 (Ω ) norm in terms of the best approximation error. Our final result is an L2 (Ω ) norm error estimate using approximation properties of plane waves to give an estimate for the order of convergence. Numerical examples are presented.

Mathematical and computational aspects of quaternary liquid mixing free energy measurement using light scattering

We provide a mathematical and computational analysis of light scattering measurement of mixing free energies of quaternary isotropic liquids. In previous work, we analyzed mathematical and experimental design considerations for the ternary mixture case [D. Ross, G. Thurston, and C. Lutzer, J. Chem. Phys. 129, 064106 (2008); C. Wahle, D. Ross, and G. Thurston, J. Chem. Phys. 137, 034201 (2012)]. Here, we review and introduce dimension-free general formulations of the fully nonlinear partial differential equation (PDE) and its linearization, a basis for applying the method to composition spaces of any dimension, in principle. With numerical analysis of the PDE as applied to the light scattering implied by a test free energy and dielectric gradient combination, we show that values of the Rayleigh ratio within the quaternary composition tetrahedron can be used to correctly reconstruct the composition dependence of the free energy. We then extend the analysis to the case of a finite number of data points, measured with noise. In this context the linearized PDE describes the relevant diffusion of information from light scattering noise to the free energy. The fully nonlinear PDE creates a special set of curves in the composition tetrahedron, collections of which form characteristics of the nonlinear and linear PDEs, and we show that the information diffusion has a time-like direction along the positive normals to these curves. With use of Monte Carlo simulations of light scattering experiments, we find that for a modest laboratory light scattering setup, about 100-200 samples and 100 s of measurement time are enough to be able to measure the mixing free energy over the entire quaternary composition tetrahedron, to within an L(2) error norm of 10(-3). The present method can help quantify thermodynamics of quaternary isotropic liquid mixtures.

Two splitting positive definite mixed finite element methods for parabolic integro-differential equations

Two novel mixed finite element procedures are established for parabolic integro-differential equations, which can be split into two independent symmetric positive definite sub-schemes and do not need to solve a coupled system of equations without requiring the LBB consistency condition. The convergence analysis shows that both methods lead to the optimal order L2(Ω) norm error estimate for u and optimal H(div;Ω) norm error estimate for σ. A numerical example is presented to illustrate the theoretical analysis.

Robust inversion using biweight norm and its application to seismic inversion

In spite of some minor drawbacks such as nonuniqueness and higher computational cost, finding the least-absolute (l1 norm) error solution to solve an optimisation problem is mostly known to give a better answer than the classical least-squares (l2 norm) method. This is because the robust property of the median value is affected little by outlier values and the solution of the least l1 norm error corresponds to the solution of minimum median error. Several variants of the l1 norm such as the Huber norm and the Hybrid norm have the same robust properties as the l1 norm. The optimisation methods based on l1 norm obtain their robustness by reducing the influence of outliers significantly, although never ignoring it. Therefore, if the proportion of outliers increases, most of the methods based on l1 norm may begin to be affected by the outliers. In such a case, other types of robust measures such as Tukey’s Biweight (Bisquare weight) norm, which excludes outliers in computing the misfit measure, could perform better. This paper describes the application of the Biweight norm using the IRLS (iteratively reweighted least-squares) method as a robust inversion and shows its possible improvement in robustness when dealing with data having many outliers.

Stability and convergence of fully discrete finite element schemes for the acoustic wave equation

In this paper, we investigate the stability and convergence of some fully discrete finite element schemes for solving the acoustic wave equation where a discontinuous Galerkin discretization in space is used. We first review and compare conventional time-stepping methods for solving the acoustic wave equation. We identify their main properties and investigate their relationship. The study includes the Newmark algorithm which has been used extensively in applications. We present a rigorous stability analysis based on the energy method and derive sharp stability results covering some well-known CFL conditions. A convergence analysis is carried out and optimal a priori error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the L2-norm error over a finite time interval converges optimally as O(hp+1+Δts), where p denotes the polynomial degree, s=1 or 2, h the mesh size, and Δt the time step.

Remapping Method for Transferring Data between Two Meshes Using a Modified Iterative SPR Approach for Parallel Resolution

In metal forming simulations, the transfer of data from one mesh to another can be very often required such as, in ALE formulation at each time step or in Lagrangian formulation at each remeshing step. Its accuracy is one of the main concerns for researchers, since the accumulation of generated diffusion can lead to larger numerical error and create convergence problems. Since 1992, the Super convergent Patch Recovery method (SPR) introduced by Zienkiewicz & Zhu [1], was a major breakthrough for the methods to estimate errors in FE solution. Later, it has also been used to recover nodal fields from integration points, in order to transfer data between two meshes [10]. The original method raises some difficulties to treat the domain boundaries. They have not quite properly been handled, in particular in the frame of parallel implementation, despite the fact that, surface phenomena, such as contact and friction, play such an important role in metal forming applications. In the present paper, it is presented a modified iterative SPR method which deals with boundary points with the same order of accuracy as the interior points. It does not require increasing the patch size and it is easier to implement in parallel environment. Also, when interpolating the field on new mesh, a new and consistent technique (P0+ transport) of enriched field has been used. It involves building a P1+ field by mixing the recovered SPR nodal field with the known field at integration points. Results are presented in form of convergence rate and L2 error norm, for several analytical functions for a SPR based transfer operator against a simple volumetric based transfer operator, before being applied to an actual metal forming problem. All computations presented here have been done by using four processors in parallel environment.

Optimal Recovery of Holomorphic Functions from Inaccurate Information about Radial Integration

This paper addresses the optimal recovery of functions from Hilbert spaces of functions on the unit disc. The estimation, or recovery, is performed from inaccurate information given by integration along radial paths. For a holomorphic function expressed as a series, three distinct situations are considered: where the information error in L2 norm is bound by δ＞0 or for a finite number of terms the error in l2N norm is bound by δ＞0 or lastly the error in the jth coefficient is bound by δj＞0. The results are applied to the Hardy-Sobolev and Bergman-Sobolev spaces.