Superconvergence Error Estimate Research Articles

Superconvergence of a new energy dissipation finite element scheme for nonlinear Schrödinger equation with wave operator

This paper is devoted to develop a new energy dissipation scheme by finite element method (FEM) for a nonlinear Schrödinger equation with wave operator (NLSW) and investigate its superconvergence behavior through introducing a new second order discrete approximation with respect to time. The new scheme not only guarantees the mass and energy stable but also is efficient in achieving the numerical solution at the last point. Also, an application to other general nonlinear equations with wave operator is discussed. On the one hand, the boundedness of the solution about the numerical equation is deduced and as a result, the existence and uniqueness of the solution are obtained. On the other hand, the superconvergence error estimate is derived and some numerical results are given to confirm the theoretical predictions.

Unconditionally superconvergent error estimates of a linearized Galerkin finite element method for the nonlinear thermistor problem

Unconditionally superconvergent error estimates of a linearized Galerkin finite element method for the nonlinear thermistor problem

Convergence and superconvergence analysis of energy-preserving Crank–Nicolson Galerkin method for the Benjamin–Bona–Mahony equation

In this paper, the convergence and superconvergence error estimates are investigated for Benjamin–Bona–Mahony equation with energy-preserving Crank–Nicolson Galerkin scheme. Firstly, a priori bound of the numerical solution is obtained due to the energy conservation property of the numerical scheme. Secondly, the existence and uniqueness of the numerical solution is proved rigorously. Then, the superclose error estimate between the numerical solution and Ritz projection of the exact solution in -norm is derived by skilfully dealing with the nonlinear term. Moreover, the superclose error estimate between numerical solution and interpolation of the exact solution in -norm is acquired with a novel approach, which is the combination of the interpolation operator and Ritz projection operator as well as high accuracy error estimate of the bilinear element. Furthermore, the superconvergence error estimate in -norm is obtained with a simple and efficient interpolation post-processing approach. Finally, some numerical results are provided to verify the theoretical analysis.

Lp error estimate of nonlinear Schrödinger equation using a two‐grid finite element method

AbstractA two‐grid finite element scheme is presented for nonlinear Schrödinger equation. superconvergence error estimates of finite element solutions based on bound of finite element solutions is proved. The optimal order error estimations of the two‐grid solution in the and norm are given without any time‐step size conditions. It is shown, both theoretically and numerically, that the coarse space can be extremely coarse, with no loss in the order of accuracy.

Superconvergence analysis of Galerkin method for semilinear parabolic integro-differential equation

In this paper, based on the bilinear element used for spatial discretization and a linearized backward Euler scheme used for temporal discretization, the superconvergence error estimate is derived for semilinear parabolic integro-differential equation without certain time-step restrictions. The key is to derive a uniform boundness of the numerical solution in energy norm under the weaker assumption compared to previous literatures for nonlinear term. Numerical results are presented to confirm the correctness of the theoretical analysis.

A linearized conservative nonconforming FEM for nonlinear Klein-Gordon-Schrödinger equations

A linearized mass and energy-conservative nonconforming FEM is proposed for the nonlinear Klein-Gordon-Schrödinger (KGS) equations. Firstly, by introducing an auxiliary time-discrete system, the error is separated into two parts, i.e., one from the temporal discretization and the other from the spatial discretization. Then, the uniform bounds of the solution to the above time-discrete system in some strong norms and the error estimates in temporal direction are derived through rigorous analysis. Secondly, superclose and superconvergence error estimates in the sense of broken H1-norm are obtained without any restrictions on the time step-size. There are two ingredients in our analysis. One is that the Lp(2≤p<∞)-norm instead of L∞-norm is utilized to bound numerical solutions. The other is that the special characters of EQ1rot element (Lemma 2) are applied to cope with the consistent errors based on the H2 regularity of the time-discrete system. Particularly, sharp error estimates are also employed to deal with the tough nonlinear terms. Finally, numerical tests are given to illustrate our theoretical results.

Superconvergence error estimates of discontinuous Galerkin time stepping for singularly perturbed parabolic problems

Superconvergence error estimates of discontinuous Galerkin time stepping for singularly perturbed parabolic problems

Unconditionally optimal convergence of an energy-conserving and linearly implicit scheme for nonlinear wave equations

In this paper, we present and analyze an energy-conserving and linearly implicit scheme for solving the nonlinear wave equations. Optimal error estimates in time and superconvergent error estimates in space are established without certain time-step restrictions. The key is to estimate directly the solution bounds in the H2-norm for both the nonlinear wave equation and the corresponding fully discrete scheme, while the previous investigations rely on the temporal-spatial error splitting approach. Numerical examples are presented to confirm energy-conserving properties, unconditional convergence and optimal error estimates, respectively, of the proposed fully discrete schemes.

How to Design a Generic Accuracy-Enhancing Filter for Discontinuous Galerkin Methods

Higher order accuracy is one of the well-known beneficial properties of the discontinuous Galerkin (DG) method. Furthermore, many studies have demonstrated the superconvergence property of the semi-discrete DG method. One can take advantage of this superconvergence property by post-processing techniques to enhance the accuracy of the DG solution. The smoothness-increasing accuracy-conserving (SIAC) filter is a popular post-processing technique introduced by Cockburn et al. (Math. Comput. 72(242): 577–606, 2003). It can raise the convergence rate of the DG solution (with a polynomial of degree k) from order \(k+1\) to order \(2k+1\) in the \(L^2\) norm. This paper first investigates general basis functions used to construct the SIAC filter for superconvergence extraction. The generic basis function framework relaxes the SIAC filter structure and provides flexibility for more intricate features, such as extra smoothness. Second, we study the distribution of the basis functions and propose a new SIAC filter called compact SIAC filter that significantly reduces the support size of the original SIAC filter while preserving (or even improving) its ability to enhance the accuracy of the DG solution. We prove the superconvergence error estimate of the new SIAC filters. Numerical results are presented to confirm the theoretical results and demonstrate the performance of the new SIAC filters.

Superconvergence error estimate of Galerkin method for Sobolev equation with Burgers' type nonlinearity

In this paper, based on the implicit Euler scheme in the temporal direction, the superconvergence property is investigated by using the special property of the bilinear element on the rectangular mesh for the Sobolev equation with Burgers' nonlinearity. The existence and uniqueness of the fully-discrete solution is proved. Further, the superconvergence error estimate in L∞(H1)-norm is established in terms of a novel approach, i.e., the technique of the combination of the interpolation operator and projection operator. Finally, a numerical experiment is carried out to confirm the theoretical analysis.

Superconvergence error estimate of a linearized energy-stable Galerkin scheme for semilinear wave equation

In this paper, a linearized energy-stable Galerkin scheme is investigated for a semilinear wave equation. Based on the special property of the bilinear element on the rectangular mesh, the superconvergence error estimate in L∞(H1(Ω)) is obtained in terms of a suitable post-processing approach. Finally, a numerical example is presented to support the theoretical analysis.

Superconvergence Error Estimate of a Finite Element Method on Nonuniform Time Meshes for Reaction–Subdiffusion Equations

In this paper, we consider superconvergence error estimates of finite element method approximation of Caputo’s time fractional reaction–subdiffusion equations under nonuniform time meshes. For the standard Galerkin method we see that the optimal order error estimate of temporal direction cannot be derived from the weak formulation of the problem. We establish a time-space error splitting argument, which are called the temporal error and the spatial error, respectively. The temporal error is proved skillfully based on an improved discrete Gronwall inequality. We obtain the sharp temporal $$H^1$$ -norm error estimates with respect to the convergence order of the approximate solution and $$H^1$$ -norm superclose results are given in details. Furthermore, by virtue of the interpolated postprocessing techniques, the global $$H^1$$ -norm superconvergence results are presented. Finally, we present some numerical results that give insight into the reliability of the theoretical analysis.

A split-step finite-element method for incompressible Navier-Stokes equations with high-order accuracy up-to the boundary

An efficient and accurate finite-element algorithm is described for the numerical solution of the incompressible Navier-Stokes (INS) equations. The new algorithm that solves the INS equations in a velocity-pressure reformulation is based on a split-step scheme in conjunction with the standard finite-element method. The split-step scheme employed for the temporal discretization of our algorithm completely separates the pressure updates from the solution of velocity variables. When the pressure equation is formed explicitly, the algorithm avoids solving a saddle-point problem; therefore, our algorithm has more flexibility in choosing finite-element spaces. In contrast, popular mixed finite-element methods that solve the INS equations in the primitive variables (or velocity-divergence formulation) lead to discrete saddle-point problems whose solution depends on the choice of finite-element spaces for velocity and pressure that is subject to the well-known Ladyzenskaja-Babuška-Brezzi (LBB) (or inf-sup) condition. For efficiency and robustness, Lagrange (piecewise-polynomial) finite elements of equal order for both velocity and pressure are used. Accurate numerical boundary condition for the pressure equation is also investigated. Motivated by a post-processing technique that calculates derivatives of a finite element solution with super-convergent error estimates, an alternative numerical boundary condition is proposed for the pressure equation at the discrete level. The new numerical pressure boundary condition that can be regarded as a better implementation of the compatibility boundary condition improves the boundary-layer errors of the pressure solution. A normal-mode analysis is performed using a simplified model problem on a uniform mesh to demonstrate the numerical properties of our methods. Convergence studies using P1 elements support the analytical results and demonstrates that our algorithm with the new numerical boundary condition achieves the optimal second-order accuracy for both velocity and pressure up-to the boundary. Benchmark problems are also computed and carefully compared with existing studies. Finally, as an example to illustrate that our approach can be easily adapted for higher-order finite elements, we solve the classical flow-past-a-cylinder problem using Pn finite elements with n≥1.

Unconditional superconvergence analysis of a linearized Crank–Nicolson Galerkin FEM for generalized Ginzburg–Landau equation

In this paper, we study an effective numerical method for the generalized Ginzburg–Landau equation (GLE). Based on the linearized Crank–Nicolson difference method in time and the standard Galerkin finite element method with bilinear element in space, the time-discrete and space–time discrete systems are both constructed. We focus on a rigorous analysis and consideration of unconditional superconvergence error estimates of the discrete schemes. Firstly, by virtue of the temporal error results, the regularity for the time-discrete system is presented. Secondly, the classical Ritz projection is used to obtain the spatial error with order O(h2) in the sense of L2-norm. Thanks to the relationship between the Ritz projection and the interpolated projection, the superclose estimate with order O(τ2+h2) in the sense of H1-norm is derived. Thirdly, it follows from the interpolated postprocessing technique that the global superconvergence result is deduced. Finally, some numerical results are provided to confirm the theoretical analysis.

High accuracy error estimates of a Galerkin finite element method for nonlinear time fractional diffusion equation

AbstractIn this work, an effective and fast finite element numerical method with high‐order accuracy is discussed for solving a nonlinear time fractional diffusion equation. A two‐level linearized finite element scheme is constructed and a temporal–spatial error splitting argument is established to split the error into two parts, that is, the temporal error and the spatial error. Based on the regularity of the time discrete system, the temporal error estimate is derived. Using the property of the Ritz projection operator, the spatial error is deduced. Unconditional superclose result in H1‐norm is obtained, with no additional regularity assumption about the exact solution of the problem considered. Then the global superconvergence error estimate is obtained through the interpolated postprocessing technique. In order to reduce storage and computation time, a fast finite element method evaluation scheme for solving the nonlinear time fractional diffusion equation is developed. To confirm the theoretical error analysis, some numerical results are provided.

Superconvergence of numerical gradient for weak Galerkin finite element methods on nonuniform Cartesian partitions in three dimensions

A superconvergence error estimate for the gradient approximation of the second order elliptic problem in three dimensions is analyzed by using weak Galerkin finite element scheme on the uniform and non-uniform cubic partitions. Due to the loss of the symmetric property from two dimensions to three dimensions, this superconvergence result in three dimensions is not a trivial extension of the recent superconvergence result in two dimensions Li et al. (0000) from rectangular partitions to cubic partitions. The error estimate for the numerical gradient in the L2-norm arrives at a superconvergence order of O(hr)(1.5≤r≤2) when the lowest order weak Galerkin finite elements consisting of piecewise linear polynomials in the interior of the elements and piecewise constants on the faces of the elements are employed. A series of numerical experiments are illustrated to confirm the established superconvergence theory in three dimensions.

Superconvergence analysis of low order nonconforming finite element methods for variational inequality problem with displacement obstacle

Superconvergence analysis of nonconforming finite element methods (FEMs) are discussed for solving the second order variational inequality problem with displacement obstacle. The elements employed have a common typical character, i.e., the consistency error can reach order O(h3/2−ɛ), nearly 1/2 order higher than their interpolation error when the exact solution of the considered problem belongs to H5/2−ɛ(Ω) for any ε > 0. By making full use of special properties of the element’s interpolations and Bramble–Hilbert lemma, the superconvergence error estimates of order O(h3/2−ɛ) in the broken H1-norm are derived. Finally, some numerical results are provided to confirm the theoretical results.

Low order nonconforming finite element method for time-dependent nonlinear Schr\xf6dinger equation

The main aim of this paper is to apply a low order nonconforming mathit{EQ}_{1}^{mathrm{rot}} finite element to solve the nonlinear Schrödinger equation. Firstly, the superclose property in the broken H^{1}-norm for a backward Euler fully-discrete scheme is studied, and the global superconvergence results are deduced with the help of the special characters of this element and the interpolation postprocessing technique. Secondly, in order to reduce computing cost, a two-grid method is developed and the corresponding superconvergence error estimates are obtained. Finally, a numerical experiment is carried out to confirm the theoretical analysis.

A Hybridizable Discontinuous Galerkin Method for Kirchhoff Plates

With the introduction of numerical traces respectively related to the normal bending moment, the twisting moment and the effective transverse shear force, and based on the Hermann–Miyoshi formulation, this paper proposes a hybridizable discontinuous Galerkin (HDG) method for Kirchhoff plate bending problems. The piecewise polynomials of degrees $$k-1$$ and k are used to approximate the moment and the deflection, respectively. The optimal and superconvergent error estimates are derived under minimal regularity assumptions on the exact solution. The key ingredients in the analysis include the derivation of a discrete inf-sup condition and some local lower bound estimates of a posteriori error analysis. The significant feature of the HDG method is superconvergence as well as the low number of globally coupled degrees of freedom associated with Lagrange multipliers. Furthermore, a new discrete deflection is constructed by postprocessing the solution of the HDG method, which superconverges to the deflection with order $$k+1$$ in broken $$H^1$$ norm. Finally, some numerical results are shown to demonstrate the theoretical results.

Superconvergence of Ritz‐Galerkin finite element approximations for second order elliptic problems

In this paper, the author derives an ‐superconvergence for the piecewise linear Ritz‐Galerkin finite element approximations for the second‐order elliptic equation equipped with Dirichlet boundary conditions. This superconvergence error estimate is established between the finite element solution and the usual Lagrange nodal point interpolation of the exact solution, and thus the superconvergence at the nodal points of each element. The result is based on a condition for the finite element partition characterized by the coefficient tensor and the usual shape functions on each element, called ‐equilateral assumption in this paper. Several examples are presented for the coefficient tensor and finite element triangulations which satisfy the conditions necessary for superconvergence. Some numerical experiments are conducted to confirm this new theory of superconvergence.