Superconvergence Analysis Research Articles

Superconvergence analysis of a low order expanded conforming mixed finite element method for the extended Fisher–Kolmogorov equation

This paper is devoted to the superconvergence analysis of a low order expanded conforming mixed finite element method(MFEM) for the extended Fisher–Kolmogorov equation. Both semi-discrete and linearized backward Euler fully-discrete schemes are analyzed with bilinear and Nédéleć elements. The present work has two main contributions: One is that some a priori bounds of the approximations are derived, which lead to the unique solvability of the semi-discrete scheme and make the nonlinear term can be dealt with efficiently. By such a way, the superclose properties and superconvergence results can be derived through the high accuracy results of the above two elements and the interpolated postprocessing technique, respectively. The other one is that a novel splitting technique is utilized to deal with the nonlinear term, which can avoid the use of popular error splitting technique and the restrictions between the time step size and mesh size. Besides, the combination technique of interpolation and projection operators also plays an important role in the superclose and superconvergence analysis. Finally, some numerical results are provided to show the validity of the theoretical analysis.

Superconvergence analysis and extrapolation of a BDF2 fully discrete scheme for nonlinear reaction–diffusion equations

Superconvergence analysis and extrapolation of a BDF2 fully discrete scheme for nonlinear reaction–diffusion equations

Superconvergence analysis of low order nonconforming finite element method for coupled nonlinear semiconductor device problem

Superconvergence analysis of low order nonconforming finite element method for coupled nonlinear semiconductor device problem

Unconditional superconvergence analysis of a new energy stable nonconforming BDF2 mixed finite element method for BBM–Burgers equation

The focus of this paper is to establish a new energy stable 2-step backward differentiation formula (BDF2) fully-discrete mixed finite element method (MFEM) for the Benjamin–Bona–Mahony–Burgers (BBM–Burgers) equation with the nonconforming rectangular EQ1rot element and zero-order Raviart–Thomas (R–T) element (EQ1rot/Q10×Q01). Based on the energy stable property, the existence and uniqueness of the numerical solution are proved by the Brouwer fixed point theorem. Subsequently, with the assistance of some typical properties of this element pair and the interpolation post-processing approach, the unconditional superclose and superconvergence results with O(h2+τ2) are obtained directly without any constraints between the spatial partition parameter h and the time step τ. It is worthy to mention that the method and analysis presented herein are very different from the time–space splitting approach utilized in the previous studies and simplify the implement. Finally, two numerical experiments are executed to validate the theoretical analysis.

Unconditionally superconvergence error analysis of an energy-stable finite element method for Schrödinger equation with cubic nonlinearity

In this paper, the unconditionally superconvergence analysis is studied for the cubic Schrödinger equation with an energy-stable finite element method. A different approach is proposed to obtain the unconditionally superclose error estimate in H1-norm firstly without using the time splitting technique required in the previous literature. The key to the analysis is to use a priori boundedness of the numerical solution in energy norm and control the nonlinear terms rigorously by two cases, i.e., τ≤h2 and τ≥h2, where τ denotes the temporal size and h is the spatial size. Subsequently, the global superconvergence error estimate in H1-norm is derived by an effective interpolation post-processing approach. Finally, some numerical experiments are carried out to confirm the theoretical findings.

Convergence and superconvergence analysis for a mass conservative, energy stable and linearized BDF2 scheme of the Poisson–Nernst–Planck equations

In this paper, we consider a linearized BDF2 finite element scheme for the Poisson–Nernst–Planck (PNP) equations. By employing a novel approach, we rigorously derive unconditional optimal error estimates of the numerical solutions in the l∞(L2) and l∞(H1) norms, as well as superconvergent results. The key of the convergence and superconvergence analysis lies in deriving the stability of the finite element solutions in some stronger norms. The advantage of this approach is that there is no need to introduce a corresponding time discrete system, so it is more concise than the error split technique in previous literatures. Finally, we carry out two numerical examples to confirm the theoretical findings.

Superconvergence analysis of the conforming discontinuous Galerkin method on a Bakhvalov-type mesh for singularly perturbed reaction–diffusion equation

The conforming discontinuous Galerkin (CDG) method maximizes the utilization of all degrees of freedom of the discontinuous Pk polynomial to achieve a convergence rate two orders higher than its counterpart conforming finite element method employing continuous Pk element. Despite this superiority, there is little theory of the CDG methods for singular perturbation problems. In this paper, superconvergence of the CDG method is studied on a Bakhvalov-type mesh for a singularly perturbed reaction–diffusion problem. For this goal, a pre-existing least squares method has been utilized to ensure better approximation properties of the projection. On the basis of that, we derive superconvergence results for the CDG finite element solution in the energy norm and L2-norm and obtain uniform convergence of the CDG method for the first time.

Superconvergence analysis of an energy-conserving Galerkin FEM for the two-dimensional sine-Gordon equation

In this paper, the superconvergence error analysis of an energy-conserving Galerkin fully discrete scheme is proposed and investigated for the two-dimensional sine-Gordon equation. The unique solvability of the numerical scheme as well as the energy conservation are studied firstly. Then, based on the special property of the bilinear element on the rectangular mesh and the superclose estimate between interpolation and Ritz projection of the exact solution in H 1 -norm, the global superconvergence result in H 1 -norm is obtained in terms of a post-processing technique. Finally, numerical results are provided to confirm the energy conservation and superconvergence properties.

Superconvergence analysis of spectral volume methods for one-dimensional diffusion and third-order wave equations

Superconvergence analysis of spectral volume methods for one-dimensional diffusion and third-order wave equations

Error estimates of the direct discontinuous Galerkin method for two-dimensional nonlinear convection–diffusion equations: Superconvergence analysis

This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for two-dimensional nonlinear convection–diffusion equations. To copy with the complexity and difficulty caused by the nonlinear term, we utilize the method of Taylor expansion and correction idea to achieve our superconvergence goal. Specifically, we first adopt Taylor expansion to decompose the error into two parts: a high-order nonlinear term and a lower-order linear term. Then by using the idea of correction function for the linear term, a high order error bound is obtained. By doing so, a unified analysis of DDG method for general nonlinear problems is finally established. We prove that, for any piecewise tensor-product polynomials of degree k≥2, the DDG solution is superconvergent at nodes and Lobatto points, with an order of O(h2k) and O(hk+2), respectively. Moreover, superconvergence properties for the derivative approximation are also studied and the superconvergence points are identified at Gauss points, with an order of O(hk+1). Numerical experiments are presented to confirm the sharpness of all the theoretical findings.

Superconvergence analysis of the nonconforming FEM for the Allen–Cahn equation with time Caputo–Hadamard derivative

This paper first introduces a high-order approximate formula for the Caputo–Hadamard fractional derivative using interpolation theory, and analyzes its convergence and accuracy. Although the formula bears resemblance to the logarithmic L2-1σ formula in form, it differs in the selection of mesh points, making it suitable for more complex models in numerical computations. Subsequently, a numerical method for the Allen–Cahn equation with time Caputo–Hadamard derivative is proposed. The temporal direction is discretized using the newly derived difference formula, while the spatial direction is approximated by the anisotropic nonconforming quasi-Wilson finite element method (FEM). It is proven that the derived scheme has optimal convergence accuracy in the L2-norm and global superconvergence property in the H1-norm. Numerical examples are provided to corroborate the theoretical claims.

Unconditional superconvergence analysis of an energy dissipation property preserving nonconforming FEM for nonlinear BBMB equation

Unconditional superconvergence analysis of an energy dissipation property preserving nonconforming FEM for nonlinear BBMB equation

A new linearized second-order energy-stable finite element scheme for the nonlinear Benjamin-Bona-Mahony-Burgers equation

In this paper, a novel linearized second-order energy-stable fully discrete scheme for the nonlinear Benjamin-Bona-Mahony-Burgers (BBMB) equation is introduced, along with a superconvergence analysis of the conforming finite element method (FEM). Through skillful decomposition of the nonlinear term u∇⋅u, a new linearized second-order fully discrete scheme is developed. This scheme requires fewer iterations and exhibits higher computational efficiency compared to the nonlinear approach. Furthermore, it maintains energy stability, enabling direct numerical solution boundedness in the H1-norm, which represents an improvement over the L∞-norm boundedness reported in previous literature. Secondly, relying on this boundedness and the high-precision results of the bilinear element, we derive the unconditional error estimates for superclose and global superconvergence without any restrictions on the time step size Δt and spatial mesh size h. Finally, numerical examples are presented to validate the theoretical analysis.

Superconvergence analysis of a two-grid BDF2-FEM for nonlinear dispersive wave equation

The aim of this paper is to study the superconvergent behavior of a two-grid finite element method (FEM) with 2-step backward differential formula (BDF2) for nonlinear dispersive wave equation. By introducing an auxiliary variable q=ut for the original variable u, the problem is transformed into a parabolic system. With the help of the combination technique of the interpolation and Ritz projection, the superclose and superconvergent estimates for the above two variables in H1-norm are obtained under the lower regularity of the exact solution compared with the standard Galerkin FEM. Finally, some numerical results are provided to show the correctness of the theoretical predictions and good performance of the proposed method.

Novel superconvergence analysis of a low order FEM for nonlinear time-fractional Joule heating problem

The aim of this paper is to develop and investigate a fully-discrete scheme with conforming P1 element for the nonlinear time-fractional Joule heating problem in which the Caputo derivative is approximated by the classical L1 method. First, a novel superclose estimate in the H1-norm is derived rigorously with some new analysis techniques under low regularity of the solutions un,ϕn∈L∞(0,T;H3(Ω)) rather than un∈L∞(0,T;H4(Ω)) and ϕn∈L∞(0,T;H3(Ω)∩W2,∞(Ω)) required in the previous studies. Then, the global superconvergence result is deduced by interpolated post-processing approach. Finally, some numerical results are provided to verify the theoretical analysis. It should be mentioned that the analysis and results presented herein are also valid to some other known conforming and nonconforming finite elements.

Pointwise superconvergence of block finite elements for the three-dimensional variable coefficient elliptic equation

<p>This study investigated the point-wise superconvergence of block finite elements for the variable coefficient elliptic equation in a regular family of rectangular partitions of the domain in three-dimensional space. Initially, the estimates for the three-dimensional discrete Greens function and discrete derivative Greens function were presented. Subsequently, employing an interpolation operator of projection type, two essential weak estimates were derived, which were crucial for superconvergence analysis. Ultimately, by combining the aforementioned estimates, we achieved superconvergence estimates for the derivatives and function values of the finite element approximation in the point-wise sense of the $ L^\infty $-norm. A numerical example illustrated the theoretical results.</p>

Superconvergence analysis of an [formula omitted]-Galerkin mixed FEM for Klein–Gordon–Zakharov equations with power law nonlinearity

This paper aims to study an H1-Galerkin mixed finite element method (FEM) for Klein–Gordon–Zakharov (KGZ) equations with power law nonlinearity. The bilinear element and zero-order Raviart–Thomas element (Q11/Q10×Q01) are taken as approximation spaces for the original variables p, ϕ and the auxiliary variables u=∇p and ψ=∇ϕ, respectively. The supercloseness and superconvergence estimates of order O(h2+τ2) are presented for p and ϕ in the H1 norm and u and ψ in the L2 norm, which improve the known results that the original variable ϕ can only reach optimal error estimate shown in numerical experiments without theoretical analysis. Here h is the subdivision parameter and τ is the time step. Finally, some numerical results are provided to support the theoretical analysis.

Unconditional superconvergence analysis of an energy-stable L1 scheme for coupled nonlinear time-fractional prey-predator equations with nonconforming finite element

The constrained nonconforming rotated Q1 (CNQ1rot) finite element is applied to develop an energy-stable L1 fully-discrete scheme through the Caputo fractional derivative approximation for the coupled nonlinear time-fractional prey-predator equations, and the unconditional superconvergence behavior is investigated. In which a novel high-accuracy estimate for the CNQ1rot element is given with the help of the Bramble-Hilbert (B-H) lemma, and the energy stability of the numerical solution is confirmed. Both of them are critical for demonstrating the unique solvability of the proposed scheme via the Brouwer fixed point theorem, and deducing the superclose estimation without any limitations between the spatial division size h and the time step τ. In addition, the above high-accuracy estimate can be extended to anisotropic meshes. Then, the unconditional superconvergence result is obtained by introducing the interpolation post-processing operator. Lastly, the theoretical findings are supported by some numerical experiments.

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.

Unconditional superconvergence analysis of an energy-preserving finite element scheme for nonlinear BBM equation

An energy-preserving Crank-Nicolson (CN) fully-discrete scheme is developed with the nonconforming Quasi-Wilson element for the nonlinear Benjamin-Bona-Mahony (BBM) equation. The existence and uniqueness of the numerical solution are demonstrated by the Brouwer fixed point theorem. Then with the help of the special character of this element, that is, the consistency error can reach order O(h2), one order higher than its interpolation error, and the interpolation post-processing technique, the unconditional supercloseness and superconvergence behavior on quadrilateral meshes are derived rigorously without the restriction between mesh size h and time step τ. Finally, some numerical experiments are carried out to confirm the theoretical analysis.