Optimal Error Analysis Research Articles

Optimal error estimates of a decoupled finite element scheme for the unsteady inductionless MHD equations

AbstractThis article focuses on a new and optimal error analysis of a decoupled finite element scheme for the inductionless magnetohydrodynamic (MHD) equations. The method uses the classical inf‐sup stable Mini/Taylor‐Hood (Mini/TH) finite element pairs to appropriate the velocity and pressure, and Raviart–Thomas (RT) face element to discretize the current density spatially, and the semi‐implicit Euler scheme with an additional stabilized term and some delicate implicit–explicit handling for the coupling terms temporally. The method enjoys some impressive features that it is linear, decoupled, unconditional energy stable and charge‐conservative. Due to the errors from the explicit handing of the coupling terms and the existence of the artificial stabilized term, and the contamination of the lower‐order RT face discretization in the error analysis, the existing theoretical results are not unconditional and optimal. By utilizing the anti‐symmetric structure of the coupling terms and the existence of the extra dissipative term, and the negative‐norm estimate for the mixed Poisson projection, we establish the unconditional and optimal error estimates for all the variables. Numerical tests are presented to illustrate our theoretical findings.

A new optimal error analysis of a mixed finite element method for advection–diffusion–reaction Brinkman flow

AbstractThis article deals with the error analysis of a Galerkin‐mixed finite element methods for the advection–reaction–diffusion Brinkman flow in porous media. Numerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest‐order Galerkin‐mixed method is the most popular one, where the linear Lagrange element is used for the concentration and the lowest‐order Raviart–Thomas element, the lowest‐order Nédélec edge element and piece‐wise constant discontinuous Galerkin element are used for the velocity, vorticity and pressure, respectively. The existing error estimate of this lowest‐order finite element method is only for all variables in spatial direction, which is not optimal for the concentration variable. This paper focuses on a new and optimal error estimate of a linearized backward Euler Galerkin‐mixed FEMs, where the second‐order accuracy for the concentration in spatial directions is established unconditionally. The key to our optimal error analysis is a new negative norm estimate for Nédélec edge element. Moreover, based on the computed numerical concentration, we propose a simple one‐step recovery technique to obtain a new numerical velocity, vorticity and pressure with second‐order accuracy. Numerical experiments are provided to confirm our theoretical analysis.

Optimal error analysis of an unconditionally stable BDF2 finite element approximation for the 3D incompressible MHD equations with variable density

This paper presents a second-order finite element scheme for the approximations of the three-dimensional (3D) incompressible magnetohydrodynamics system with variable density (VD-MHD), where two-step backward differentiation formula (BDF2) is used to discrete the time derivative and the (P2,P1b,P1,P1) finite elements are used to approximate the density, velocity, pressure and magnetic. Based on an equivalent VD-MHD system, the proposed finite element algorithm is unconditionally stable in the sense that the discrete energy inequalities hold without small condition of the time step size. After a rigorous analysis, the optimal L2 error estimates (τ2+h2) of the density, velocity field and magnetic field are established, where τ and h are the time step size and mesh size. Finally, numerical results are given to support these convergence rates.

Optimal \(\boldsymbol{{L^2}}\) Error Estimates of Unconditionally Stable Finite Element Schemes for the Cahn–Hilliard–Navier–Stokes System

The paper is concerned with the analysis of a popular convex-splitting finite element method (FEM) for the Cahn–Hilliard–Navier–Stokes system, which has been widely used in practice. Since the method is based on a combined approximation to multiple variables involved in the system, the approximation to one of the variables may seriously affect the accuracy for others. Optimal-order error analysis for such combined approximations is challenging. The previous works failed to present optimal error analysis in -norm due to the weakness of the traditional approach. Here we first present an optimal error estimate in -norm for the convex-splitting FEMs. We also show that optimal error estimates in the traditional (interpolation) sense may not always hold for all components in the coupled system due to the nature of the pollution/influence from lower-order approximations. Our analysis is based on two newly introduced elliptic quasi-projections and the superconvergence of negative norm estimates for the corresponding projection errors. Numerical examples are also presented to illustrate our theoretical results. More important is that our approach can be extended to many other FEMs and other strongly coupled phase field models to obtain optimal error estimates.

A novel approach of unconditional optimal error estimate of linearized and conservative Galerkin FEM for Klein–Gordon–Schrödinger equations

This paper is devoted to the unconditional optimal error analysis of a linearized, decoupled and conservative Galerkin finite element method (FEM) for the Klein–Gordon–Schödinger equations. The present work has two main contributions: One is that the mass and energy conservation properties of the scheme are proved, and a priori boundedness of the numerical solutions is derived. The other one is that a novel approach is developed and analyzed, which can avoid the use of popular error splitting technique and the restrictions between the time step size and mesh size required in the previous works, and may lead to the unconditional optimal error estimates for the related variables. Some numerical results are also provided to confirm the theoretical analysis and show the efficiency of the proposed scheme.

Optimal error analysis of space–time second-order difference scheme for semi-linear non-local Sobolev-type equations with weakly singular kernel

In this paper, we construct and analyze a Crank–Nicolson difference scheme for solving the semilinear Sobolev-type equation with the Riemann–Liouville fractional integral of order α∈(0,1). The proposed scheme consists of two main stages. First, to compensate for the singular behavior of the exact solution at the initial time t=0, a Crank–Nicolson scheme and the product trapezoidal integration rule are proposed on graded meshes for time discretization. Second, a classical finite difference operator on uniform meshes is used for space discretization. Under suitable assumptions of the regularity condition, the stability and convergence analysis of this scheme in a new norm are given by the energy argument. It is shown that how the mesh grading exponent γ affects the temporal convergence rate of the presented scheme in the final convergence result. This scheme can attain a second-order accuracy in both time and space by choosing an optimal grading parameter γ. Numerical results confirm that the error analysis is sharp, and comparisons with results on uniform temporal meshes are included to indicate the effectiveness of our method.

Positive definite multi-kernels for scattered data interpolations

In this article, we use the knowledge of positive definite tensors to develop a concept of positive definite multi-kernels to construct the kernel-based interpolants of scattered data. By the techniques of reproducing kernel Banach spaces, the optimal recoveries and error analysis of the kernel-based interpolants are shown for a special class of strictly positive definite multi-kernels.

Unconditionally Optimal Error Analysis of the Second-Order BDF Finite Element Method for the Kuramoto-Tsuzuki Equation

Unconditionally Optimal Error Analysis of the Second-Order BDF Finite Element Method for the Kuramoto-Tsuzuki Equation

Optimal L2 error analysis of first-order Euler linearized finite element scheme for the 2D magnetohydrodynamics system with variable density

In this paper, a first-order Euler semi-implicit finite element scheme is proposed for solving the two-dimensional incompressible magnetohydrodynamics flows with variable density numerically. The proposed FEM algorithm is unconditionally stable at the full discrete level, which is a key issue in designing efficient algorithms for the multi-physical field problems. When the time step size τ and the mesh size h both are sufficiently small, optimal spatial error estimates in L 2 -norm are presented for the density, velocity and magnetic field by using energy techniques based on uniform regularities of solutions to the time-discrete scheme and the method of mathematical induction. Finally, numerical results are given to confirm the theoretical analysis.

Closed-loop deep neural network optimal control algorithm and error analysis for powered landing under uncertainties

Closed-loop deep neural network optimal control algorithm and error analysis for powered landing under uncertainties

Error analysis of a fully discrete projection method for magnetohydrodynamic system

AbstractIn this paper, we develop and analyze a finite element projection method for magnetohydrodynamics equations in Lipschitz domain. A fully discrete scheme based on Euler semi‐implicit method is proposed, in which continuous elements are used to approximate the Navier–Stokes equations and H(curl) conforming Nédélec edge elements are used to approximate the magnetic equation. One key point of the projection method is to be compatible with two different spaces for calculating velocity, which leads one to obtain the pressure by solving a Poisson equation. The results show that the proposed projection scheme meets a discrete energy stability. In addition, with the help of a proper regularity hypothesis for the exact solution, this paper provides a rigorous optimal error analysis of velocity, pressure and magnetic induction. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme.

Optimal error analysis of the spectral element method for the 2D homogeneous wave equation

Optimal a priori error bounds are theoretically derived, and numerically verified, for approximate solutions to the 2D homogeneous wave equation obtained by the spectral element method. To be precise, the spectral element method studied here takes advantage of the Gauss-Lobatto-Legendre quadrature, thus resulting in under-integrated elements but a diagonal mass matrix. The approximation error in H1 is shown to be of order O(hp) with respect to the element size h and of order O(p−q) with respect to the degree p, where q is the spatial regularity of the solution. These results improve on past estimates in the L2 norm, particularly with respect to h. Specific assumptions on the discretization by the spectral element method are the use of a triangulation by quadrilaterals constructed via affine transformations from a reference square element and of a second order discretization in time by the leap-frog scheme.

Optimal error analysis of the Alikhanov formula for a time-fractional Schrödinger equation

Optimal error analysis of the Alikhanov formula for a time-fractional Schrödinger equation

Unconditional stability and optimal error analysis of mass conservative characteristic mixed FEM for wormhole propagation

This paper is concerned with unconditional stability and optimal convergence of mass conservative type method for simulating wormhole propagation in porous media. Specifically, mass conservative characteristic finite element method (FEM) is used for the solute transport equation, the mixed FEM is used for velocity-pressure equation and Galerkin FEM for porosity equation. By error splitting technique, we prove the error of the solution between full discrete system and time discrete system is time-independent, while the numerical solution is bounded without certain time step restriction. Moreover, the optimal L2 error estimates further hold in a general case by elliptic quasi-projection, where the unconditional r+1 order accuracy of the concentration and porosity can be obtained with no loss of accuracy for r order approximation velocity-pressure equation. Numerical experiments are presented to verify the theoretical analysis and the effectiveness of the proposed method.

Mixed virtual volume methods for elliptic problems

We develop a class of mixed virtual volume methods for elliptic problems on polygonal/polyhedral grids. Unlike the mixed virtual element methods introduced in [22,13], our methods are reduced to symmetric, positive definite problems for the primary variable without using Lagrangian multipliers. We start from the usual way of changing the given equation into a mixed system using the Darcy's law, u=−K∇p. By integrating the system of equations with some judiciously chosen test spaces on each element, we define new mixed virtual volume methods of all orders. We show that these new schemes are equivalent to the nonconforming virtual element methods for the primal variable p.Once the primary variable is computed solving the symmetric, positive definite system, all the degrees of freedom for the Darcy velocity are locally computed. Also, the L2-projection onto the polynomial space is easy to compute. Hence our work opens an easy way to compute Darcy velocity on the polygonal/polyhedral grids. For the lowest order case, we give a formula to compute a Raviart-Thomas space like representation which satisfies the conservation law.An optimal error analysis is carried out and numerical results are presented which support the theory.

An [formula omitted]-Galerkin mixed finite element method for identification of time dependent parameters in parabolic problems

A direct method of identification of time dependent parameters in a linear parabolic boundary value problem with over-specified total internal energy involves the flux at the boundary, and an H1 mixed formulation seems to be more suitable than the standard methods for such class of nonlocal problems. Therefore, this paper develops and analyses an H1-Galerkin mixed finite element method. Optimal error estimates in both primary and flux variables are derived in semidiscrete case. Moreover, a priori error estimate for the parameters is established. Based on linearised backward Euler method, a completely discrete scheme is proposed and optimal error analysis is derived. The results of the numerical experiments show the efficacy of the proposed method and confirm our theoretical results.

The maximum norm analysis of a nonmatching grids method for a class of parabolic $p(x)$-Laplacien equation

Motivated by the work of Boulaaras and Haiour in [7], we provide a maximum norm analysis of Schwarz alternating method for parabolic p(x)-Laplacien equation, where an optimal error analysis each subdomain between the discrete Schwarz sequence and the continuous solution of the presented problem is established

Unconditionally Optimal Error Analysis of a Linear Euler FEM Scheme for the Navier–Stokes Equations with Mass Diffusion

Unconditionally Optimal Error Analysis of a Linear Euler FEM Scheme for the Navier–Stokes Equations with Mass Diffusion

On a three step two‐grid finite element method for the Oldroyd model of order one

AbstractIn this work, an optimal error analysis of a three step two‐grid method for the equations of motion arising in the 2DOldroyd model of order one is discussed. The model, which can be thought of as an integral perturbation of Navier‐Stokes equations (NSE), represents linear viscoelastic fluid flows. This non‐linear model is analyzed here using a three step numerical scheme. In the first step the problem is solved on a coarse grid, and we use this course grid solution to linearize the problem and solve it in second step, on a finer grid. Third step is a correcting step done on the finer grid. Optimal error estimates for the velocity in and ‐norms and for the pressure in ‐norm in the semidiscrete case are established. Estimates are shown to be uniform under the uniqueness assumption. Then, based on backward Euler method, a completely discrete scheme is analyzed and optimal a priori error estimates are derived. All the analysis are carried out for the non‐smooth initial data. Finally we present some numerical results to validate our theoretical results. These examples show that the three step two‐grid method is efficient than solving a nonlinear problem directly, as is expected.

Optimal Error Analysis of Euler and Crank--Nicolson Projection Finite Difference Schemes for Landau--Lifshitz Equation

The Landau--Lifshitz equation has been widely used to describe the dynamics of magnetization in a ferromagnetic material, which is highly nonlinear with the nonconvex constraint $|{m}|=1$. A crucial issue in designing efficient numerical schemes is to preserve this constraint in the discrete level. A simple and frequently used one is the projection method, which projects the numerical solution onto a unit sphere at each time step. The method has been used in many areas in the past several decades, while analysis has not been explored. In this paper, we present optimal error analysis of a backward Euler and a Crank--Nicolson semi-implicit projection finite difference scheme for the Landau--Lifshitz equation. The analysis is based on new and precise estimates of the difference between the errors of projected and unprojected solutions in both $L^2$ and $H^1$ norms. Some numerical experiments are provided to confirm our theoretical results.