Superclose Results Research Articles

Superconvergent analysis of a nonconforming mixed finite element method for non-stationary conduction–convection problem

In this paper, a backward-Euler mixed finite element (MFE) approximation scheme is presented for the non-stationary conduction–convection problem, in which the constrained nonconforming rotated (CNR) Q1 element is used to approximate the velocity u, the temperature T and the Q0 element to the pressure p. Based on the characters of these elements and some special skills, i.e., mean-value skill and a new transforming skill with respect to τ, the superclose results of order O(h2+τ) for u,T in the broken H1-norm and p in L2-norm are deduced, respectively. Here, h is the subdivision parameter and τ, the time step. Furthermore, the global superconvergent results are obtained through the interpolation postprocessing technique. Finally, numerical results are provided to confirm the theoretical analysis.

An efficient nonconforming finite element two-grid method for Allen–Cahn equation

In this paper, superconvergent analysis of an efficient two-grid method is discussed for the Allen–Cahn equation with the nonconforming EQ1rot finite element. The unconditional stability of the numerical scheme is proved based on the monotonically increasing character of the nonlinear term in the problem, while the previous works always require some certain stability conditions. By use of the typical properties of EQ1rot element together with a more accurate estimate on the nonlinear term, the superclose result of order O(h2+H4+τ) in the broken H1-norm is deduced rigorously without the above restrictions for the first time. Furthermore, the global superconvergence behavior is derived through interpolated postprocessing skill. Numerical results illustrate that the proposed method is actually effective.

A priori and a posteriori estimates of stabilized mixed finite volume methods for the incompressible flow arising in arteriosclerosis

The flow arising in arteriosclerosis is modeled by the incompressible flow with a slip boundary condition of friction type in this paper, whose weak solution satisfies a variational inequality. We develop and analyze the stabilized mixed finite volume methods for the resulting model. We first achieve the same optimal order results of the finite volume methods as those of the corresponding finite element methods under the same regularity on the exact solution and a slightly additional regularity on the source term. Also, a super-close result between the respective solutions of the finite element methods and finite volume methods is derived for the first time. For adaptivity, we further derive a posterior estimate for the present problem satisfying the variational inequality, which reduces the computational cost substantially and provides efficient error control for the solution. Finally, numerical results are shown to support the developed convergence theory.

Uniform superconvergent analysis of a new mixed finite element method for nonlinear Bi-wave singular perturbation problem

Uniform superconvergent analysis of a new low order nonconforming mixed finite element method (MFEM) is studied for solving the fourth order nonlinear Bi-wave singular perturbation problem (SPP) by EQ1rot element. On one hand, the existence, uniqueness and stability of the numerical solutions are proved. On the other hand, with the help of the special characters of this element, uniform superclose result of order O(h2) for the original variable in the broken H1 norm and uniform optimal order estimate of order O(h2) for the intermediate variable in L2 norm are deduced irrelevant to the real perturbation parameter δ appearing in the considered problem, respectively. In which, the nonlinear term in the Bi-wave SPP, which is the main difficulty in the whole error analysis, is dealt with rigorously through a novel splitting technique. Moreover, the global uniform superconvergent estimate is obtained with the interpolated postprocessing approach. Finally, some numerical results are provided to confirm the theoretical analysis. Here h is the subdivision parameter.

Uniform superconvergence analysis of Ciarlet‐Raviart scheme for Bi‐wave singular perturbation problem

Uniform superconvergence analysis of the Ciarlet‐Raviart mixed finite element scheme is discussed for solving the fourth‐order Bi‐wave singular perturbation problem (SPP) by the bilinear element. Firstly, the existence and uniqueness of the approximation solution are proved. Secondly, with the help of the special characters of this element, uniform superclose result of order O(h2) for the original variable in H1 norm and uniform optimal order estimate of order O(h2) for the intermediate variable in L2 norm are deduced with respect to the real perturbation parameter δ appearing in the considered problem. Furthermore, the global uniform superconvergent estimate is obtained through the interpolated postprocessing approach. Finally, some numerical results are provided to verify the theoretical analysis. Here, h denotes the mesh size.

Nonconforming quasi-Wilson finite element method for 2D multi-term time fractional diffusion-wave equation on regular and anisotropic meshes

The paper mainly focuses on studying nonconforming quasi-Wilson finite element fully-discrete approximation for two dimensional (2D) multi-term time fractional diffusion-wave equation (TFDWE) on regular and anisotropic meshes. Firstly, based on the Crank–Nicolson scheme in conjunction with L1-approximation of the time Caputo derivative of order α ∈ (1, 2), a fully-discrete scheme for 2D multi-term TFDWE is established. And then, the approximation scheme is rigorously proved to be unconditionally stable via processing fractional derivative skillfully. Moreover, the superclose result in broken H1-norm is deduced by utilizing special properties of quasi-Wilson element. In the meantime, the global superconvergence in broken H1-norm is derived by means of interpolation postprocessing technique. Finally, some numerical results illustrate the correctness of theoretical analysis on both regular and anisotropic meshes.

Fast evaluation and high accuracy finite element approximation for the time fractional subdiffusion equation

In this article, an efficient algorithm for the evaluation of the Caputo fractional derivative and the superconvergence property of fully discrete finite element approximation for the time fractional subdiffusion equation are considered. First, the space semidiscrete finite element approximation scheme for the constant coefficient problem is derived and supercloseness result is proved. The time discretization is based on the L1‐type formula, whereas the space discretization is done using, the fully discrete scheme is developed. Under some regularity assumptions, the superconvergence estimate is proposed and analyzed. Then, extension to the case of variable coefficients is also discussed. To reduce the computational cost, the fast evaluation scheme of the Caputo fractional derivative to solve the fractional diffusion equations is designed. Finally, numerical experiments are presented to support the theoretical results.

Supercloseness of Primal-Dual Galerkin Approximations for Second Order Elliptic Problems

We show that two widely used Galerkin formulations for second-order elliptic problems provide approximations which are actually superclose, that is, their difference converges faster than the corresponding errors. In the framework of linear elasticity, the two formulations correspond to using either the stiffness tensor or its inverse the compliance tensor. We find sufficient conditions, for a wide class of methods (including mixed and discontinuous Galerkin methods), which guarantee a supercloseness result. For example, for the HDG $$_{k}$$ method using polynomial approximations of degree $${k>0}$$ , we find that the difference of approximate fluxes superconverges with order $${k+2}$$ and that the difference of the scalar approximations superconverges with order $${k+3}$$ . We provide numerical results verifying our theoretical results.

Galerkin finite element methods for convection–diffusion problems with exponential layers on Shishkin triangular meshes and hybrid meshes

In this work, we provide a convergence analysis for a Galerkin finite element method on a Shishkin triangular mesh and a hybrid mesh for a singularly perturbed convection–diffusion equation. The hybrid mesh replaces the triangles of the Shishkin mesh by rectangles in the layer regions. The supercloseness results are established that the computed solution converges to the interpolant of the true solution with 3/2 order and 2 order (up to a logarithmic factor) on the two kinds of mesh, respectively. These convergence rates are uniformly valid with respect to the diffusion parameter and imply that the hybrid mesh is superior to the Shishkin triangular mesh. Numerical experiments illustrate these theoretical results.

Supercloseness of continuous interior penalty method for convection–diffusion problems with characteristic layers

A singularly perturbed convection–diffusion problem posed on the unit square is solved using a continuous interior penalty (CIP) method with piecewise bilinears on a rectangular Shishkin mesh. A detailed analysis proves a new stability bound for the CIP method, in a norm that is stronger than the usual CIP norm. This bound enables a new supercloseness result for the CIP method: the computed solution is shown to be second order (up to a logarithmic factor) convergent in the new strong norm to the piecewise bilinear interpolant of the true solution. As a corollary one obtains almost optimal order convergence in the L2 norm of the CIP solution to the true solution. Numerical experiments illustrate these theoretical results.

Unconditional superconvergence analysis for nonlinear hyperbolic equation with nonconforming finite element

Nonlinear hyperbolic equation is studied by developing a linearized Galerkin finite element method (FEM) with nonconforming EQ1rot element. A time-discrete system is established to split the error into two parts which are called the temporal error and the spatial error, respectively. The temporal error is proved skillfully which leads to the analysis for the regularity of the time-discrete system. The spatial error is derived τ-independently with order O(h2+hτ) in broken H1-norm. The final unconditional superclose result of u with order O(h2+τ2) is deduced based on the above achievements. The two typical characters of this nonconforming EQ1rot element (see Lemma 1 below) play an important role in the procedure of proof. At last, a numerical example is provided to support the theoretical analysis. Here, h is the subdivision parameter, and τ, the time step.

Unconditional superconvergence analysis of conforming finite element for nonlinear parabolic equation

Galerkin finite element approximation to nonlinear parabolic equation is studied with a linearized backward Euler scheme. The error between the exact solution and the numerical solution is split into two parts which are called the temporal error and the spatial error through building a time-discrete system. On one hand, the temporal error derived skillfully leads to the regularity of the time-discrete system solution. On the other hand, the τ-independent spatial error and the boundedness of the numerical solution in L∞-norm is deduced with the above achievements. At last, the superclose result of order O(h2+τ) in H1-norm is obtained without any restriction of τ in a routine way. Here, h is the subdivision parameter, and τ, the time step.

Superconvergence analysis of an [formula omitted]-Galerkin mixed finite element method for Sobolev equations

An H1-Galerkin mixed finite element method (MFEM) is discussed for the Sobolev equations with the bilinear element and zero order Raviart–Thomas element (Q11+Q10×Q01). The existence and uniqueness of the solutions about the approximation scheme are proved. Two new important lemmas are given by using the properties of the integral identity and the Bramble–Hilbert lemma, which lead to the superclose results of order O(h2) for original variable u in H1 norm and flux q→ in H(div;Ω) norm under semi-discrete scheme. Furthermore, two new interpolated postprocessing operators are put forward and the corresponding global superconvergence results are obtained. On the other hand, a second order fully-discrete scheme with superclose property O(h2+τ2) is also proposed. At last, numerical experiment is included to illustrate the feasibility of the proposed method. Here h is the subdivision parameter and τ is the time step.

Unconditional superconvergence analysis of a new mixed finite element method for nonlinear Sobolev equation

In this paper, a new mixed finite element scheme is proposed for the nonlinear Sobolev equation by employing the finite element pair Q11/Q01 × Q10. Based on the combination of interpolation and projection skill as well as the mean-value technique, the τ-independent superclose results of the original variable u in H1-norm and the variable q→=−(a(u)∇ut+b(u)∇u) in L2-norm are deduced for the semi-discrete and linearized fully-discrete systems (τ is the temporal partition parameter). What’s more, the new interpolated postprocessing operators are put forward and the corresponding global superconvergence results are obtained unconditionally, while previous literature always require certain time step conditions. Finally, some numerical results are provided to confirm our theoretical analysis, and show the efficiency of the method.

Supercloseness Result of Higher Order FEM/LDG Coupled Method for Solving Singularly Perturbed Problem on S-Type Mesh

we present a first supercloseness analysis for higher order FEM/LDG coupled method for solving singularly perturbed convection-diffusion problem. Based on piecewise polynomial approximations of degreek (k≥1), a supercloseness property ofk+1/2in DG norm is established on S-type mesh. Numerical experiments complement the theoretical results.

A new low order least squares nonconforming characteristics mixed finite element method For Burgers’ equation

In this paper, a new low order least squares nonconforming characteristics mixed finite element method (MFEM) is considered for two-dimensional Burgers’ equation. By use of two typical characters of the elements for approximating the velocity and flux variables: (a) the consistency errors of the elements can be estimated as order O(h2) with respect to the mesh size h in the broken H1-norm, one order higher than their interpolation errors; (b) the elements’ interpolation operators satisfy certain orthogonalities, which lead to O(h2) order error estimates in L2-norm and some superclose results, the accuracy of corresponding variables in L2-norm is improved by one order compared with other MFEMs in the existing literature. It seems that the results provided herein have never been derived before.

High accuracy analysis of the characteristic‐nonconforming FEM for a convection‐dominated transport problem

A low order characteristic‐nonconforming finite element method is proposed for solving a two‐dimensional convection‐dominated transport problem. On the basis of the distinguish property of element, that is, the consistency error can be estimated as order O(h2), one order higher than that of its interpolation error, the superclose result in broken energy norm is derived for the fully discrete scheme. In the process, we use the interpolation operator instead of the so‐called elliptic projection, which is an indispensable tool in the traditional finite element analysis. Furthermore, the global superconvergence is obtained by using the interpolated postprocessing technique. Lastly, some numerical experiments are provided to verify our theoretical analysis. Copyright © 2013 John Wiley & Sons, Ltd.

Nonconforming quadrilateral finite element method for a class of nonlinear sine–Gordon equations

Nonconforming quadrilateral finite element method (FEM) of the two-dimensional nonlinear sine–Gordon equation is studied for semi-discrete and Crank–Nicolson fully-discrete schemes, respectively. Firstly, we prove a special feature of a new arbitrary quadrilateral element (named modified Quasi–Wilson element), i.e., the consistency error is of order O(h2) (h denotes the mesh size) in H1-norm, which leads to optimal order error estimate and superclose result with order O(h2) for the semi-discrete scheme through a different approach from the existing literature. Secondly, because the consistency error estimate of the new modified Quasi–Wilson element can reach a staggering O(h3) order, two orders higher than that of interpolation error, the optimal order error estimates of Crank–Nicolson fully-discrete scheme are obtained on arbitrary quadrilateral meshes with Ritz projection. Moreover, a superclose result in H1-norm is presented on generalized rectangular meshes by a new technique. Thirdly, the global superconvergence results of H1-norm for both semi-discrete and fully-discrete schemes are derived on rectangular meshes with interpolated postprocessing technique. Finally, a numerical test is carried out to verify the theoretical analysis.

EQ 1 rot nonconforming finite element method for nonlinear dual phase lagging heat conduction equations

EQ1rot nonconforming finite element approximation to a class of nonlinear dual phase lagging heat conduction equations is discussed for semi-discrete and fully-discrete schemes. By use of a special property, that is, the consistency error of this element is of order O(h2) one order higher than its interpolation error O(h), the superclose results of order O(h2) in broken H1-norm are obtained. At the same time, the global superconvergence in broken H1-norm is deduced by interpolation postprocessing technique. Moreover, the extrapolation result with order O(h4) is derived by constructing a new interpolation postprocessing operator and extrapolation scheme based on the known asymptotic expansion formulas of EQ1rot element. Finally, optimal error estimate is gained for a proposed fully-discrete scheme by different approaches from the previous literature.

Supercloseness on graded meshes for [formula omitted] finite element approximation of a reaction–diffusion equation

In this paper we analyze the standard piece-wise bilinear finite element approximation of a model reaction–diffusion problem. We prove supercloseness results when appropriate graded meshes are used. The meshes are those introduced in Durán and Lombardi (2005) [8] but with a stronger restriction on the graduation parameter. As a consequence we obtain almost optimal error estimates in the L2-norm thus completing the error analysis given in Durán and Lombardi (2005) [8].