Superconvergence Analysis Research Articles

Unconditional superconvergence analysis of modified finite difference streamlined diffusion method for nonlinear convection-dominated diffusion equation

In this article, a Crank-Nicolson fully discrete finite element scheme of modified finite difference streamlined diffusion (MFDSD) method is developed and investigated for nonlinear convection-dominated diffusion equation, which can get rid of the numerical oscillation appeared in Galerkin finite element method (FEM). The supercloseness and superconvergence estimates of order O(h2+τ2) in H1 norm are derived without the restriction between the time step τ and the mesh size h. Firstly, a time discrete system is established to split the error into two parts - the temporal error and spatial error, and the regularity of the solution of the time discrete system is deduced with the help of mathematical induction. Then the numerical solution is bounded in L∞ norm by the spatial error which leads to the above unconditional supercloseness property, and the global superconvergence result is deduced through interpolation post-processing technique. Lastly, two numerical examples are provided to verify the correctness of the theoretical analysis and to show the big advantage of the proposed MFDSD method over the Galerkin FEM.

Superconvergence Analysis of Discontinuous Galerkin Methods for Systems of Second-Order Boundary Value Problems

In this paper, we present an innovative approach to solve a system of boundary value problems (BVPs), using the newly developed discontinuous Galerkin (DG) method, which eliminates the need for auxiliary variables. This work is the first in a series of papers on DG methods applied to partial differential equations (PDEs). By consecutively applying the DG method to each space variable of the PDE using the method of lines, we transform the problem into a system of ordinary differential equations (ODEs). We investigate the convergence criteria of the DG method on systems of ODEs and generalize the error analysis to PDEs. Our analysis demonstrates that the DG error’s leading term is determined by a combination of specific Jacobi polynomials in each element. Thus, we prove that DG solutions are superconvergent at the roots of these polynomials, with an order of convergence of O(hp+2).

Unconditional superconvergence analysis of an energy conservation scheme with Galerkin FEM for nonlinear Benjamin–Bona–Mahony equation

In this paper, a general energy conservation Crank–Nicolson (CN) fully-discrete finite element method (FEM) scheme is developed for solving the nonlinear Benjamin–Bona–Mahony (BBM) equation, and the unconditional supercloseness and superconvergence behavior are discussed with the nonconforming EQ1rot element. Different from the time–space splitting technique, the boundedness of numerical solution in the broken H1-norm is obtained directly based on the above energy conservation property, and the well-posedness of numerical solution is proved rigorously through the Brouwer fixed point theorem. Then, with the help of the special characters of this element and the interpolation post-processing technique, the unconditional superclose and superconvergence results are deduced strictly without any restriction between mesh size h and time step τ. Further, our analysis is extended to some other popular conforming and nonconforming finite elements. Finally, two numerical experiments are implemented to confirm the theoretical analysis. It is worth mentioning that this paper not only improve the previous results, but also can be regard as a framework for solving the BBM equation and some other PDEs with Galerkin FEMs.

Unconditionally convergent and superconvergent analysis of second-order weighted IMEX FEMs for nonlinear Ginzburg-Landau equation

Unconditionally convergent and superconvergent analysis of second-order weighted IMEX FEMs for nonlinear Ginzburg-Landau equation

Convergence and superconvergence analysis of finite element methods for nonlinear Ginzburg–Landau equation with Caputo derivative

Convergence and superconvergence analysis of finite element methods for nonlinear Ginzburg–Landau equation with Caputo derivative

Unconditional superconvergence analysis of a modified nonconforming energy stable BDF2 FEM for Sobolev equations with Burgers’ type nonlinearity

The aim of this paper is to develop a modified energy stable 2-step backward differentiation formula (BDF2) fully discrete scheme and study its superconvergence behavior with the nonconforming quadrilateral quasi-Wilson element for the Sobolev equations with Burgers’ type nonlinearity. The existence and uniqueness of the numerical solution are proved by Brouwer fixed point theorem. Then, different from the so-called time–space splitting technique, some typical properties of this element especially the consistency error can reach order O(h2), one order higher than its interpolation error, and the interpolation post-processing approach are used to derive the unconditional supercloseness and superconvergence results on quadrilateral meshes without the restriction between the mesh size h and the time step τ. At last, a numerical example is carried out and a comparison with the Wilson element is also given to confirm the theoretical analysis.

Superconvergence analysis of the bilinear‐constant scheme for two‐dimensional incompressible convective Brinkman–Forchheimer equations

AbstractIn this article, a low order conforming mixed finite element method is proposed and investigated for two‐dimensional convective Brinkman–Forchheimer equations. Based on the special properties of the bilinear‐constant finite element pair on the rectangular mesh and the careful treatment of the nonlinear terms, the superclose error estimates for velocity in ‐norm and pressure in ‐norm are obtained. Then, in terms of interpolation post‐processing technique, the global superconvergence results are derived. Finally, some numerical experiments are carried out to demonstrate the correctness of the theoretical findings.

Super convergence analysis of fully discrete Hermite splines to simulate wave behaviour of Kuramoto–Sivashinsky equation

An improved collocation technique has been proposed to discretize the fourth-order multi-parameter non-linear Kuramoto -Sivashinsky (K-S) equation. The spatial direction has been discretized with quintic Hermite splines, whereas the temporal direction has been discretized with a weighted finite difference scheme. The fourth-order equation in space direction has been decomposed into second-order using space splitting by introducing a new variable. Space splitting has been proposed to improve the convergence of the approximate solution. The proposed equation has been analyzed on a uniform grid in both space and time directions. Error bounds for general order Hermite splines are established for fully discrete scheme. Stability analysis for the proposed scheme has also been discussed elaborately. Periodic and non periodic problems of K-S equation type have been discussed to study the technique. The error growth has been addressed by computing L2− norm and L∞− norm.

Conservative EQ1rot nonconforming FEM for nonlinear Schrödinger equation with wave operator

AbstractIn this paper, we consider leap‐frog finite element methods with element for the nonlinear Schrödinger equation with wave operator. We propose that both the continuous and discrete systems can keep mass and energy conservation. In addition, we focus on the unconditional superconvergence analysis of the numerical scheme, the key of which is the time‐space error splitting technique. The spatial error is derived independently with order in ‐norm, where and denote the space and time step size. Then the unconditional optimal error and superclose result with order are deduced, and the unconditional optimal error is obtained with order by using interpolation theory. The final unconditional superconvergence result with order is derived by the interpolation postprocessing technique. Furthermore, we apply the proposed leap‐frog finite element methods to solve the logarithmic Schrödinger equation with wave operator by introducing a regularized system with a small regularization parameter . The detailed theoretical conclusions, including the mass and energy conservation laws of the continuous regularized and discrete regularized systems and the convergence and superconvergence results, are presented as prolongation of the previous work. At last, some numerical experiments are given to confirm our theoretical analysis.

Formula omitted]-conforming virtual element method for the Laplacian eigenvalue problem in mixed form

In this paper, we propose the Stokes-type virtual element method for the Laplacian eigenvalue problem in mixed form, in which the eigenvalue problem is reformulated in terms of the velocity and the pressure of the Darcy flow. The virtual element method is H1-conforming and divergence-free. By applying these properties, the inf-sup condition and the spectral approximation theory of compact operator, we derive the optimal a priori error estimates. Then we focus on the a posteriori error estimates. Based on the superconvergence analysis, the Helmholtz decomposition and the properties of bubble functions, we prove the upper and lower bounds of the residual-type a posteriori error estimator with respect to the approximation error. Finally some numerical results are given to validate the theoretical analysis and present the good performance of the adaptive algorithm.

Unconditional superconvergence analysis of an energy-stable C–N fully discrete scheme for the nonlinear magnetic diffusion model with memory

The unconditional superconvergence behavior of an energy-stable Crank–Nicolson (C–N) fully discrete scheme with nonconforming EQ1rot finite element method (FEM) is investigated for the nonlinear integro-differential system associated with the diffusion of the magnetic field. The stability of the numerical solution in the broken H1-norm is proved by the Gagliardo–Nirenberg (G–N) inequality, and the unique solvability is demonstrated by the Brouwer fixed point theorem. In which the boundness of the L∞-norm and the inverse inequality are eliminated. The combination of the Green formula and two special properties of EQ1rot element leads to the unconditional superconvergence estimate without any restriction between the time step and space partition size. Finally, the validity of the theoretical findings is supported by some numerical results.

Superconvergence Analysis of $C^m$ Finite Element Methods for Fourth-Order Elliptic Equations I: One Dimensional Case

Superconvergence Analysis of $C^m$ Finite Element Methods for Fourth-Order Elliptic Equations I: One Dimensional Case

A super-convergence analysis of the Poisson solver with Octree grids and irregular domains

Resolving the difficulty of T-junctions in Octree grids, Losasso et al. [9] introduced an ingenious Poisson solver with rectangular domains and Neumann boundary conditions. Its numerical solution was empirically observed [9] to be second order convergent and its numerical gradient was rigorously proved [8] to be one and a half order convergent, which is the so-called super-convergence.This article is devoted to extending the Poisson solver and its supporting proof from rectangular to irregular domains. The generalized Whitney decomposition [12] efficiently generates octree grids for irregular domains imposing the finest resolution near the boundary of domain. Combined with the Heaviside treatment [17,4], the Poisson solver is extended to irregular domains and a novel and rigorous analysis shows that the aforementioned super-convergence still holds true.

Superconvergence analysis of finite element method for nonlinear semiconductor device problem

Superconvergence analysis of finite element method for nonlinear semiconductor device problem

Superconvergence analysis of a linearized energy-conservative Galerkin method for the nonlinear Schrödinger equation with wave operator

Superconvergence analysis of a linearized energy-conservative Galerkin method for the nonlinear Schrödinger equation with wave operator

Nodal Accuracy Improvement Technique for Linear Elements with Application to Adaptivity

In the finite element method, the conventional linear elements have long been precluded, due to their low accuracy of nodal displacements, from the analysis of super-convergence and adaptivity via the element energy projection (EEP) technique. To overcome this problem, in this paper, a nodal accuracy improvement technique is proposed for linear elements in 1D to 3D problems. In this method, a residual nodal load vector is derived with the conventional EEP solution, and a simple back-substitution process can generate the improved nodal displacements without changing the global stiffness matrix. Subsequently, an improved EEP scheme for linear elements is proposed based on the improved nodal displacements. Finally, by using the improved EEP solution as an error estimator, a two-phased adaptive algorithm is presented. Numerical examples show that the accuracy of nodal displacements is improved from the second-order convergence to the fourth-order convergence by using the nodal accuracy improvement technique, and the EEP solutions for element interior displacements are improved from the second-order convergence to the third-order convergence by using the improved EEP scheme. Therefore, the improved EEP scheme can be effectively used as an error estimator in adaptivity analysis for linear elements, which turns out to be efficient in general and even outperforms cubic elements for singularity problems.

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.

Superconvergence analysis of nonconforming finite element method for two-dimensional time-fractional Allen–Cahn equation

In this paper, we investigate the numerical solution of the time-fractional Allen–Cahn equation with variable diffusion coefficients. A fully discrete approximation is presented by the L2-1σ scheme on graded meshes and spatial high precision nonconforming finite element method (FEM). The nonlinear term is treated with the Newton linearization method. Based on the modified discrete fractional Grönwall inequality, we rigorously prove that the proposed scheme can achieve optimal convergence accuracy in both time and space directions. Furthermore, the H1-norm global superconvergence result is obtained through the interpolation postprocessing technique. Finally, numerical experiments confirm the correctness of our theoretical analysis.

Superconvergence analysis of a conservative mixed finite element method for the nonlinear Klein–Gordon–Schrödinger equations

AbstractIn this paper, a linearized mass and energy conservative mixed finite element method (MFEM) is proposed for solving the nonlinear Klein–Gordon–Schrödinger equations. Optimal error estimates without grid‐ratio condition are derived by some rigorous analysis and an error splitting technique, that is, one is the temporal error which is only τ‐dependent and the other is the spatial error which is only h‐dependent. Furthermore, the superclose results are obtained by using the idea of combination of interpolation and projection. Besides, a so‐called “lifting” approach also play an important role to obtain the superclose results. With the above achievements, the global superconvergent properties are deduced through the interpolated post processing operators. Finally, three numerical examples are given to validate the convergence order, unconditional stability, mass, and energy conservation.

Unconditional Superconvergence Analysis of Energy Conserving Finite Element Methods for the Nonlinear Coupled Klein-Gordon Equations

Unconditional Superconvergence Analysis of Energy Conserving Finite Element Methods for the Nonlinear Coupled Klein-Gordon Equations