Order Superconvergence Research Articles

High-accuracy finite element method for 2D time fractional diffusion-wave equation on anisotropic meshes

ABSTRACTEmploying finite element method in spatial direction and Crank–Nicolson scheme in temporal direction, a fully discrete scheme with high accuracy is established for a class of two-dimensional time fractional diffusion-wave equation with Caputo fractional derivative. Unconditional stability analysis of the approximate scheme is proposed. The spatial global superconvergence and temporal convergence of order for the original variable in -norm is presented by means of properties of bilinear element and interpolation postprocessing technique without Ritz projection, where h and τ are the step sizes in space and time, respectively. Finally, several numerical results are implemented to evaluate the efficiency of the theoretical results on both regular and anisotropic meshes.

Nodal superconvergence of the local discontinuous Galerkin method for singularly perturbed problems

Abstract In this paper, a superconvergence of order ( ln N ∕ N ) 2 k + 1 for the numerical traces of the LDG approximation to a one dimensional singularly perturbed convection–diffusion–reaction problem is proved. The LDG method is applied on a Shishkin mesh with 2 N elements, and we use polynomials of degree at most k on each element. This result puts the numerical finding reported in Xie and Zhang (2007), Xie et al. (2009) on firm mathematical ground.

Superconvergence of Immersed Finite Volume Methods for One-Dimensional Interface Problems

In this paper, we introduce a class of high order immersed finite volume methods (IFVM) for one-dimensional interface problems. We show the optimal convergence of IFVM in H1 and L2 norms. We also prove some superconvergence results of IFVM. To be more precise, the IFVM solution is superconvergent of order p+2 at the roots of generalized Lobatto polynomials, and the flux is superconvergent of order p+1 at generalized Gauss points on each element including the interface element. Furthermore, for diffusion interface problems, the convergence rates for IFVM solution at the mesh points and the flux at generalized Gaussian points can both be raised to 2p. These superconvergence results are consistent with those for the standard finite volume methods. Numerical examples are provided to confirm our theoretical analysis.

A $$C^0$$ Linear Finite Element Method for Biharmonic Problems

In this paper, a $$C^0$$ linear finite element method for biharmonic equations is constructed and analyzed. In our construction, the popular post-processing gradient recovery operators are used to calculate approximately the second order partial derivatives of a $$C^0$$ linear finite element function which do not exist in traditional meaning. The proposed scheme is straightforward and simple. More importantly, it is shown that the numerical solution of the proposed method converges to the exact one with optimal orders both under $$L^2$$ and discrete $$H^2$$ norms, while the recovered numerical gradient converges to the exact one with a superconvergence order. Some novel properties of gradient recovery operators are discovered in the analysis of our method. In several numerical experiments, our theoretical findings are verified and a comparison of the proposed method with the nonconforming Morley element and $$C^0$$ interior penalty method is given.

Divided difference estimates and accuracy enhancement of discontinuous Galerkin methods for nonlinear symmetric systems of hyperbolic conservation laws

In this paper, we investigate the accuracy-enhancement for the discontinuous Galerkin (DG) method for solving one-dimensional nonlinear symmetric systems of hyperbolic conservation laws. For nonlinear equations, the divided difference estimate is an important tool that allows for superconvergence of the post-processed solutions in the local L2-norm. Therefore, we first prove that the L2-norm of the α-th order (1≤ α≤ k+1) divided difference of the DG error with upwind fluxes is of order k+(3-α)/2, provided that the flux Jacobian matrix, f'(u), is symmetric positive definite. Furthermore, using the duality argument, we are able to derive superconvergence estimates of order 2k+(3-α)/2 for the negative-order norm, indicating that some particular compact kernels can be used to extract at least (3k/2+1)-th order superconvergence for nonlinear systems of conservation laws. Numerical experiments are shown to demonstrate the theoretical results.

Superconvergence of Discontinuous Galerkin methods based on upwind-biased fluxes for 1D linear hyperbolic equations

In this paper, we study superconvergence properties of the discontinuous Galerkin method using upwind-biased numerical fluxes for one-dimensional linear hyperbolic equations. A (2 k + 1)th order superconvergence rate of the DG approximation at the numerical fluxes and for the cell average is obtained under quasi-uniform meshes and some suitable initial discretization, when piecewise polynomials of degree k are used. Furthermore, surprisingly, we find that the derivative and function value approximation of the DG solution are superconvergent at a class of special points, with an order k + 1 and k + 2, respectively. These superconvergent points can be regarded as the generalized Radau points. All theoretical findings are confirmed by numerical experiments.

Superconvergence of Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives

In this paper, we present a unified approach to study superconvergence behavior of the local discontinuous Galerkin (LDG) method for high-order time-dependent partial differential equations. We select the third and fourth order equations as our models to demonstrate this approach and the main idea. Superconvergence results for the solution itself and the auxiliary variables are established. To be more precise, we first prove that, for any polynomial of degree k, the errors of numerical fluxes at nodes and for the cell averages are superconvergent under some suitable initial discretization, with an order of $$O(h^{2k+1})$$ . We then prove that the LDG solution is $$(k+2)$$ -th order superconvergent towards a particular projection of the exact solution and the auxiliary variables. As byproducts, we obtain a $$(k+1)$$ -th and $$(k+2)$$ -th order superconvergence rate for the derivative and function value approximation separately at a class of Radau points. Moreover, for the auxiliary variables, we, for the first time, prove that the convergence rate of the derivative error at the interior Radau points can reach as high as $$k+2$$ . Numerical experiments demonstrate that most of our error estimates are optimal, i.e., the error bounds are sharp.

Superconvergence of Local Discontinuous Galerkin Method for One-Dimensional Linear Schrödinger Equations

In this paper, we study the superconvergence properties of the LDG method for the one-dimensional linear Schrodinger equation. We build a special interpolation function by constructing a correction function, and prove the numerical solution is superclose to the interpolation function in the $$L^{2}$$ norm. The order of superconvergence is $$2k+1$$ , when the polynomials of degree at most k are used. Even though the linear Schrodinger equation involves only second order spatial derivative, it is actually a wave equation because of the coefficient i. It is not coercive and there is no control on the derivative for later time based on the initial condition of the solution itself, as for the parabolic case. In our analysis, the special correction functions and special initial conditions are required, which are the main differences from the linear parabolic equations. We also rigorously prove a $$(2k+1)$$ -th order superconvergence rate for the domain, cell averages, and the numerical fluxes at the nodes in the maximal and average norm. Furthermore, we prove the function value and the derivative approximation are superconvergent with a rate of $$(k+2)$$ -th order at the Radau points. All theoretical findings are confirmed by numerical experiments.

Stability and Superconvergence of MAC Scheme for Stokes Equations on Nonuniform Grids

The marker and cell (MAC) method, a class of finite volume schemes based on staggered grids, has been one of the simplest and most effective numerical schemes for solving the Stokes and Navier--Stokes equations. Its numerical superconvergence on uniform grids has been observed since 1992. In this paper we establish the LBB condition and the stability for both velocity and pressure for the MAC scheme of stationary Stokes equations on nonuniform grids. Then we construct an auxiliary function depending on the velocity and discretizing parameters and analyze the superconvengence. We obtain the second order superconvergence in the L2 norm for both velocity and pressure for the MAC scheme. We also obtain the second order superconvergence for some terms of the H1 norm of the velocity, and the other terms of the H1 norm are second order superconvergent on uniform grids. Our analysis can be extended to three dimensional problems. Numerical experiments using the MAC scheme show agreement of the numerical results with theoretical analysis.

Strong superconvergence of the Euler–Maruyama method for linear stochastic Volterra integral equations

The Euler–Maruyama method is presented for linear stochastic Volterra integral equations. Then the strong convergence property is analyzed for convolution kernels and general kernels, respectively. It is well known that for stochastic ordinary differential equations, the strong convergence order of the Euler–Maruyama method is 12. However, the strong superconvergence order of 1 is obtained for linear stochastic Volterra integral equations with convolution kernels if the kernel K2 of the diffusion term satisfies K2(0)=0; and this strong superconvergence property is inherited by linear stochastic Volterra integral equations with general kernels if the kernel K2 of the diffusion term satisfies K2(t,t)=0. The theoretical results are illustrated by extensive numerical examples.

Optimal Superconvergence of Energy Conserving Local Discontinuous Galerkin Methods for Wave Equations

AbstractThis paper is concerned with numerical solutions of the LDG method for 1D wave equations. Superconvergence and energy conserving properties have been studied. We first study the superconvergence phenomenon for linear problems when alternating fluxes are used. We prove that, under some proper initial discretization, the numerical trace of the LDG approximation at nodes, as well as the cell average, converge with an order 2k+1. In addition, we establishk+2-th order andk+1-th order superconvergence rates for the function value error and the derivative error at Radau points, respectively. As a byproduct, we prove that the LDG solution is superconvergent with an orderk+2 towards the Radau projection of the exact solution. Numerical experiments demonstrate that in most cases, our error estimates are optimal, i.e., the error bounds are sharp. In the second part, we propose a fully discrete numerical scheme that conserves the discrete energy. Due to the energy conserving property, after long time integration, our method still stays accurate when applied to nonlinear Klein-Gordon and Sine-Gordon equations.

Collocation methods for Volterra functional integral equations with non-vanishing delays

In this paper the existence, uniqueness, regularity properties, and in particular, the local representation of solutions for general Volterra functional integral equations with non-vanishing delays, are investigated. Based on the solution representation, we detailedly analyze the attainable (global and local) convergence order of (iterated) collocation solutions on θ-invariant meshes. It turns out that collocation at the m Gauss (-Legendre) points neither leads to the optimal global convergence order m+1, nor yields the local convergence order 2m on the whole interval, which is in sharp contrast to the case of the classical Volterra delay integral equations. However, if the collocation is based on the m Radau II points, the local superconvergence order 2m−1 will exhibit at all mesh points. Finally, some numerical experiments are performed to confirm our theoretical findings.

Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws: divided difference estimates and accuracy enhancement

In this paper, an analysis of the accuracy-enhancement for the discontinuous Galerkin (DG) method applied to one-dimensional scalar nonlinear hyperbolic conservation laws is carried out. This requires analyzing the divided difference of the errors for the DG solution. We therefore first prove that the alpha -th order (1 le alpha le {k+1}) divided difference of the DG error in the L^2 norm is of order {k + frac{3}{2} - frac{alpha }{2}} when upwind fluxes are used, under the condition that |f'(u)| possesses a uniform positive lower bound. By the duality argument, we then derive superconvergence results of order {2k + frac{3}{2} - frac{alpha }{2}} in the negative-order norm, demonstrating that it is possible to extend the Smoothness-Increasing Accuracy-Conserving filter to nonlinear conservation laws to obtain at least ({frac{3}{2}k+1})th order superconvergence for post-processed solutions. As a by-product, for variable coefficient hyperbolic equations, we provide an explicit proof for optimal convergence results of order {k+1} in the L^2 norm for the divided differences of DG errors and thus ({2k+1})th order superconvergence in negative-order norm holds. Numerical experiments are given that confirm the theoretical results.

The Highest Superconvergence Analysis of ADG Method for Two Point Boundary Values Problem

In this paper, an averaging discontinuous Galerkin (ADG) method for two point boundary value problems is analyzed. We prove, for any even polynomial degree k, the numerical flux convergence at a rate of $$2k+2$$2k+2 for all mesh nodes (in particular, the numerical flux for $$k=0$$k=0 has the second order superconvergence rate). Numerical experiments are shown to demonstrate the theoretical results.

Superconvergence of the direct discontinuous Galerkin method for convection‐diffusion equations

This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for one‐dimensional linear convection‐diffusion equations. We prove, under some suitable choice of numerical fluxes and initial discretization, a 2k‐th and ‐th order superconvergence rate of the DDG approximation at nodes and Lobatto points, respectively, and a ‐th order of the derivative approximation at Gauss points, where k is the polynomial degree. Moreover, we also prove that the DDG solution is superconvergent with an order k + 2 to a particular projection of the exact solution. Numerical experiments are presented to validate the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 290–317, 2017

Superconvergence of three dimensional Morley elements on cuboid meshes for biharmonic equations

In the present paper, superconvergence of second order, after an appropriate postprocessing, is achieved for three dimensional first order cuboid Morley elements of biharmonic equations. The analysis is dependent on superconvergence of second order for the consistency error and a corrected canonical interpolation operator, which help to establish supercloseness of second order for the corrected canonical interpolation. Then the final superconvergence is derived by a standard postprocessing. For first order nonconforming finite element methods of three dimensional fourth order elliptic problems, it is the first time that full superconvergence of second order is obtained without an extra boundary condition imposed on exact solutions. It is also the first time that superconvergence is established for nonconforming finite element methods of three dimensional fourth order elliptic problems. Numerical results are presented to demonstrate the validity of the theoretical results.

A class of implicit peer methods for stiff systems

We present a class of s-stage implicit two step peer methods for the solution of stiff differential equations using the function values from the previous step in addition to the new function values. This allows to increase the order to p=s and to ensure zero-stability straightforwardly. Corresponding s-stage methods for s≤6 of order p=s with optimal zero stability are presented and their stability is discussed. Under special conditions, we prove that an optimally zero-stable subclass of these methods is superconvergent of order p=s+1 for variable step sizes. Numerical tests and comparison with ode15s show the high potential of this class of implicit peer methods.

Superconvergence in collocation methods for Volterra integral equations with vanishing delays

In this paper, we investigate the optimal (global and local) convergence orders of the (iterated) collocation solutions for second-kind Volterra integral equations with vanishing delays on quasi-geometric meshes. It turns out that the classical global convergence results still hold under certain regularity conditions of the given functions. In particular, it is shown that the optimal local superconvergence order p=2m can be attained if collocation is at the m Gauss(–Legendre) points, which contrasts with collocations both on uniform meshes and on geometric meshes. Numerical experiments are performed to confirm our theoretical results.

The optimal error estimate and superconvergence of the local discontinuous Galerkin methods for one-dimensional linear fifth order time dependent equations

In this paper, we investigate the optimal error estimate and the superconvergence of linear fifth order time dependent equations. We prove that the local discontinuous Galerkin (LDG) solution is (k+1)th order convergent when the piecewise Pk space is used. Also, the numerical solution is (k+32)th order superconvergent to a particular projection of the exact solution. The numerical experiences indicate that the order of the superconvergence is (k+2), which implies the result obtained in this paper is suboptimal.

Convergence and superconvergence of a fully-discrete scheme for multi-term time fractional diffusion equations

Using finite element method in spatial direction and classical L1 approximation in temporal direction, a fully-discrete scheme is established for a class of two-dimensional multi-term time fractional diffusion equations with Caputo fractional derivatives. The stability analysis of the approximate scheme is proposed. The spatial global superconvergence and temporal convergence of order O(h2+τ2−α) for the original variable in H1-norm is presented by means of properties of bilinear element and interpolation postprocessing technique, where h and τ are the step sizes in space and time, respectively. Finally, several numerical examples are implemented to evaluate the efficiency of the theoretical results.