Superclose Results Research Articles

Superconvergence analysis of an anisotropic nonconforming FEM for parabolic equation

In this paper, the constrained nonconforming rotated Q1 (CNQ1rot) element is employed to study the superconvergence behavior of the backward-Euler (B-E) and Crank–Nicolson (C–N) fully-discrete schemes for the parabolic equation. By applying the mean-value and integral identities techniques, a novel high-accuracy estimate for the CNQ1rot element is proved on anisotropic meshes rigorously, which is essential to derive the superclose result. Then, the superconvergence results for the above two schemes are deduced with the help of the interpolation post-processing approach, respectively. Eventually, some numerical experiments are carried out to verify the rationality of the theoretical analysis.

Nonconforming mixed finite element method for the diffuse interface model of two-phase fluid flows

In this work, we consider a nonconforming mixed finite element method (MFEM) for a parabolic system consisting of the Cahn-Hilliard equations and the Navier-Stokes equations, which describes a diffuse interface model for the flow of two incompressible and immiscible fluids. We focus on the superconvergence analysis for the corresponding first order convex splitting scheme, where the bilinear element is used to approximate the Cahn-Hilliard equation in space, and the constrained nonconforming rotated (CNR) Q1 element together with the Q0 constant element to the Navier-Stokes equations. By use of the high accuracy properties of the elements and some specific techniques, the superclose results of order O(h2+τ) for the concentration ϕ and the chemical potential ω in the H1-norm, the velocity u in the broken H1-norm and the pressure p in L2-norm are strictly proved. Here, h is the subdivision parameter and τ, the time step. It is worth to say that we derive a new high accuracy character through the integral identity technique for the chosen elements, which plays a quite significant role in getting the final error results. Furthermore, the global superconvergence results of the corresponding variables are concluded by the interpolated postprocessing technique. Numerical tests are provided to justify the performance of the theoretical analysis.

Supercloseness analysis of the linear finite element method for a singularly perturbed convection–diffusion problem on Vulanović–Bakhvalov mesh

Vulanović–Bakhvalov mesh is a kind of Bakhvalov-type mesh, which has the advantages of good numerical convergence and the transition point between coarse and fine mesh is not affected by mesh parameters. However, there is little theory of finite element methods for this mesh. In this paper, a linear finite element method on a Vulanović–Bakhvalov mesh is studied for a singularly perturbed convection–diffusion problem. In this process, some properties of the Vulanović–Bakhvalov mesh are obtained. Based on these properties and a new Lagrange interpolation, we further derive the supercloseness result of 3/2th order in an energy norm.

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.

Convergence and superconvergence analysis for nonlinear delay reaction–diffusion system with nonconforming finite element

AbstractIn this article, we propose and analyze several numerical methods for the nonlinear delay reaction–diffusion system with smooth and nonsmooth solutions, by using Quasi‐Wilson nonconforming finite element methods in space and finite difference methods (including uniform and nonuniform L1 and L2‐1σ schemes) in time. The optimal convergence results in the senses of L2‐norm and broken H1‐norm, and H1‐norm superclose results are derived by virtue of two types of fractional Grönwall inequalities. Then, the interpolation postprocessing technique is used to establish the superconvergence results. Moreover, to improve computational efficiency, fast algorithms by using sum‐of‐exponential technique are built for above proposed numerical schemes. Finally, we present some numerical experiments to confirm the theoretical correctness and show the effectiveness of the fast algorithms.

On the convergence and superconvergence for a class of two‐dimensional time fractional reaction‐subdiffusion equations

AbstractThis paper analyzes a class of two‐dimensional (2‐D) time fractional reaction‐subdiffusion equations with variable coefficients. The high‐order L2‐1σ time‐stepping scheme on graded meshes is presented to deal with the weak singularity at the initial time t = 0, and the bilinear finite element method (FEM) on anisotropic meshes is used for spatial discretization. Using the modified discrete fractional Grönwall inequality, and combining the interpolation operator and the projection operator, the L2‐norm error estimation and H1‐norm superclose results are rigorously proved. The superconvergence result in the H1‐norm is derived by applying the interpolation postprocessing technique. Finally, numerical examples are presented to verify the validation of our theoretical analysis.

A two-grid $ P_0^2 $-$ P_1 $ mixed finite element scheme for semilinear elliptic optimal control problems

<abstract><p>This paper aims to construct a two-grid mixed finite element scheme for distributed optimal control governed by semilinear elliptic equations. The state and co-state are approximated by the $ P_0^2 $-$ P_1 $ pair and the control variable is approximated by the piecewise constant functions. First, a superclose result for the control variable and a priori error estimates for all variables are obtained. Second, a two-grid $ P_0^2 $-$ P_1 $ mixed finite element algorithm is presented and the corresponding error is analyzed. In the two-grid scheme, the solution of the semilinear elliptic optimal control problem on a fine grid is reduced to the solution of the semilinear elliptic optimal control problem on a much coarser grid and the solution of a linear decoupled algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. We find that the two-grid method achieves the same convergence property as the $ P_0^2 $-$ P_1 $ mixed finite element method if the two mesh sizes satisfy $ h = H^2 $. Finally, a numerical example demonstrating our theoretical results is presented.</p></abstract>

Supercloseness in a balanced norm of finite element methods on Shishkin and Bakhvalov–Shishkin rectangular meshes for reaction–diffusion problems

We present a convergence analysis for finite element methods of any order, which are applied on Shishkin mesh and Bakhvalov–Shishkin mesh to a singularly perturbed reaction–diffusion problem. A new interpolant is introduced for analysis in the balanced norm. By means of this interpolant and some superconvergence estimations, we prove supercloseness results in the cases of Shishkin mesh and Bakhvalov–Shishkin mesh. Numerical experiments also support these theoretical results.

Supercloseness of finite element method on a Bakhvalov-type mesh for a singularly perturbed problem with two parameters

In this paper, the linear finite element method on a Bakhvalov-type mesh is applied to a singularly perturbed problem with two parameters. There is a strong exponential boundary layer and a weak exponential boundary layer in the solution of this problem. In the analysis, we discuss three cases and obtain the corresponding uniform convergence and supercloseness results by using different analysis techniques. Numerical tests confirm our theoretical results.

Novel superconvergence analysis of anisotropic triangular FEM for a multi‐term time‐fractional mixed sub‐diffusion and diffusion‐wave equation with variable coefficients

AbstractA fully discrete scheme is proposed for multi‐term time‐fractional mixed sub‐diffusion and diffusion‐wave equation with variable coefficients on anisotropic meshes, where linear triangular finite elelment method (FEM) is used for the spatial discretization and modified L1 approximation coupled with Crank–Nicolson scheme is applied to temporal direction. The mixed equation concerned contains a time–space coupled derivative which is very different from the previous literature. The stability is firstly obtained. Based on the property of the projection operator, the special relation between the projection operator and the interpolation operator of linear triangular finite element, the optimal error estimation and the superclose result are deduced. Then the global superconvergence property is derived by the interpolated postprocessing technique. Some numerical experiments are carried out to confirm the theoretical analysis on anisotropic meshes.

Convergence and Supercloseness in a Balanced Norm of Finite Element Methods on Bakhvalov-Type Meshes for Reaction-Diffusion Problems

In convergence analysis of finite element methods for singularly perturbed reaction–diffusion problems, balanced norms have been successfully introduced to replace standard energy norms so that layers can be captured. In this article, we focus on the convergence analysis in a balanced norm on Bakhvalov-type rectangular meshes. In order to achieve our goal, a novel interpolation operator, which consists of a local $$L^2$$ projection operator and the Lagrange interpolation operator, is introduced for a convergence analysis of optimal order in the balanced norm. The analysis also depends on the stabilities of the $$L^2$$ projection and the characteristics of Bakhvalov-type meshes. Furthermore, we obtain a supercloseness result in the balanced norm, which appears in the literature for the first time. This result depends on another novel interpolant, which consists of the local $$L^2$$ projection operator, a vertices-edges-element operator and some corrections on the boundary.

Superconvergence analysis of nonconforming EQ1rot element for nonlinear parabolic integro‐differential equation

In this paper, superconvergent analysis for nonlinear parabolic integro‐differential equation is studied with nonconforming element. A linearized Galerkin finite element method is introduced, and a time‐discrete system is constructed to split the error into two parts. On the one hand, a rigorous analysis for the regularity of the time‐discrete system is presented based on the temporal error estimation skillfully. On the other hand, the spatial error is deduced τ‐independently with the above achievements. With the help of typical characters of this element, the superclose result of order in the broken H1‐norm is derived without any restriction of τ. At last, numerical results are provided to confirm the theoretical analysis. Here, h is the subdivision parameter, and τ is the time step.

Superconvergence of the Local Discontinuous Galerkin Method for One Dimensional Nonlinear Convection-Diffusion Equations

In this paper, we study superconvergence properties of the local discontinuous Galerkin (LDG) methods for solving nonlinear convection-diffusion equations in one space dimension. The main technicality is an elaborate estimate to terms involving projection errors. By introducing a new projection and constructing some correction functions, we prove the $$(2k+1)$$ th order superconvergence for the cell averages and the numerical flux in the discrete $$L^2$$ norm with polynomials of degree $$k\ge 1$$ , no matter whether the flow direction $$f'(u)$$ changes or not. Superconvergence of order $$k +2$$ ( $$k +1$$ ) is obtained for the LDG error (its derivative) at interior right (left) Radau points, and the convergence order for the error derivative at Radau points can be improved to $$k+2$$ when the direction of the flow doesn’t change. Finally, a supercloseness result of order $$k+2$$ towards a special Gauss–Radau projection of the exact solution is shown. The superconvergence analysis can be extended to the generalized numerical fluxes and the mixed boundary conditions. All theoretical findings are confirmed by numerical experiments.

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.

Supercloseness of linear finite element method on Bakhvalov-type meshes for singularly perturbed convection–diffusion equation in 1D

In this article, a singularly perturbed convection–diffusion equation is solved by a linear finite element method on Bakhvalov-type meshes. By means of a novel interpolation to the solution, a supercloseness result is obtained on Bakhvalov-type meshes for the first time.

Finite element error estimates in non-energy norms for the two-dimensional scalar Signorini problem

This paper is concerned with error estimates for the piecewise linear finite element approximation of the two-dimensional scalar Signorini problem on a convex polygonal domain varOmega . Using a Céa-type lemma, a supercloseness result, and a non-standard duality argument, we prove W^{1,p}(varOmega )-, L^infty (varOmega )-, W^{1,infty }(varOmega )-, and H^{1/2}(partial varOmega )-error estimates under reasonable assumptions on the regularity of the exact solution and L^p(varOmega )-error estimates under comparatively mild assumptions on the involved contact sets. The obtained orders of convergence turn out to be optimal for problems with essentially bounded right-hand sides. Our results are accompanied by numerical experiments which confirm the theoretical findings.

Convergence and superconvergence analysis of finite element methods for the time fractional diffusion equation

In this paper, two classes of finite element methods (FEMs) for the two-dimensional time fractional diffusion equation (TFDE) with non-smooth solution are proposed and analyzed. In the temporal direction, we adopt nonuniform L2-1σ method, and in the spatial direction, the conforming bilinear element and the nonconforming modified quasi-Wilson element are utilized. For the proposed conforming and nonconforming FEMs, by using new fractional discrete Grönwall inequalities, the theoretical analysis including the L2-norm error estimates and H1-norm superclose results are given in details. Furthermore, by virtue of the interpolated postprocessing techniques, the global H1-norm superconvergence results are presented. Finally, some numerical results illustrate the correctness of theoretical analysis.

Superconvergent Recovery of Rectangular Edge Finite Element Approximation by Local Symmetry Projection

A new recovery method of rectangular edge finite element approximation for Maxwell’s equations is proposed by using the local symmetry projection. The recovery method is applied to the Nedelec interpolation to obtain the superconvergence of postprocessed Nedelec interpolation. Combining with the superclose result between the Nedelec interpolation and edge finite element approximation, it is shown that the postprocessed edge finite element solution superconverges to the exact solution. Numerical examples are presented to illustrate our 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.

Unconditional superconvergent analysis of a linearized finite element method for Ginzburg-Landau equation

In this paper, unconditional superconvergent estimate with a Galerkin finite element method (FEM) is presented for Ginzburg-Landau equation by conforming bilinear FE, while all previous works require certain time-step restrictions. First of all, a time-discrete system is introduced, with which the error function is split into a temporal error and a spatial error. On one hand, the regularity of the time-discrete system is deduced with a rigorous analysis. On the other hand, the error between the numerical solution and the solution of the time-discrete system is derived τ-independently with order O(h2+hτ), where h is the subdivision parameter and τ, the time step. Then, the unconditional superclose result of order O(h2+τ) in the H1-norm is deduced directly based on the above estimates. Furthermore, the global superconvergent result is obtained through the interpolated postprocessing technique. At last, numerical results are provided to confirm the theoretical analysis.