A SUPERCONVERGENT ESTIMATE OF WILSON-LIKE ELEMENTS

In this paper, the supercouvergence of a class of Wilson-like elements is considered, and a superconvergent estimate of Wilson-like elements is obtained for strong regular meshes.

Three-Dimensional Finite Element Superconvergent Gradient Recovery on Par6 Patterns

In this paper, we present a theoretical analysis for linear finite element superconvergent gradient recovery on Par6 mesh, the dual of which is centroidal Voronoi tessellations with the lowest energy per unit volume and is the congruent cell predicted by the three-dimensional Gersho＇s conjecture. We show that the linear finite element solution uh and the linear interpolation uI have superclose gradient on Par6 meshes. Consequently, the gradient recovered from the finite element solution by using the superconvergence patch recovery method is superconvergent to Vu. A numerical example is presented to verify the theoretical result.

Superconvergent Interpolatory HDG Methods for Reaction Difusion Equations II:HHO‑Inspired Methods

In Chen et al.(J.Sci.Comput.81(3):2188–2212,2019),we considered a superconvergent hybridizable discontinuous Galerkin(HDG)method,defned on simplicial meshes,for scalar reaction-difusion equations and showed how to defne an interpolatory version which maintained its convergence properties.The interpolatory approach uses a locally postprocessed approximate solution to evaluate the nonlinear term,and assembles all HDG matrices once before the time integration leading to a reduction in computational cost.The resulting method displays a superconvergent rate for the solution for polynomial degree k≥1.In this work,we take advantage of the link found between the HDG and the hybrid high-order(HHO)methods,in Cockburn et al.(ESAIM Math.Model.Numer.Anal.50(3):635–650,2016)and extend this idea to the new,HHO-inspired HDG methods,defned on meshes made of general polyhedral elements,uncovered therein.For meshes made of shape-regular polyhedral elements afne-equivalent to a fnite number of reference elements,we prove that the resulting interpolatory HDG methods converge at the same rate as for the linear elliptic problems.Hence,we obtain superconvergent methods for k≥0 by some methods.We thus maintain the superconvergence properties of the original methods.We present numerical results to illustrate the convergence theory.

UNCONDITIONAL SUPERCONVERGENT ANALYSIS OF QUASI-WILSON ELEMENT FOR BENJAMIN-BONA-MAHONEY EQUATION

This article aims to study the unconditional superconvergent behavior of nonconforming quadrilateral quasi-Wilson element for nonlinear Benjamin Bona Mahoney(BBM)equation.For the generalized rectangular meshes including rectangular mesh,deformed rectangular mesh and piecewise deformed rectangular mesh,by use of the special character of this element,that is,the conforming part(bilinear element)has high accuracy estimates on the generalized rectangular meshes and the consistency error can reach order O(h^(2)),one order higher than its interpolation error,the superconvergent estimates with respect to mesh size h are obtained in the broken H^(1)-norm for the semi-/fully-discrete schemes.A striking ingredient is that the restrictions between mesh size h and time stepτrequired in the previous works are removed.Finally,some numerical results are provided to confirm the theoretical analysis.

NATURAL SUPERCONVERGENT POINTS OF EQUILATERAL TRIANGULAR FINITE ELEMENTS - A NUMERICAL EXAMPLE

A numerical test case demonstrates that the Lobatto and the Gauss points are not natural superconvergent points of the cubic and the quartic finite elements under equilateral triangular mesh for the Poisson equation.

New Superconvergent Structures with Optional Superconvergent Points for the Finite Volume Element Method

New superconvergent structures are proposed and analyzed for the finite volume element(FVE)method over tensorial meshes in general dimension d(for d≥2);we call these orthogonal superconvergent structures.In this framework,one has the freedom to choose the superconvergent points of tensorial k-order FVE schemes(for k≥3).This flexibility contrasts with the superconvergent points(such as Gauss points and Lobatto points)for current FE schemes and FVE schemes,which are fixed.The orthogonality condition and the modified M-decomposition(MMD)technique that are developed over tensorial meshes help in the construction of proper superclose functions for the FVE solutions and in ensuring the new superconvergence properties of the FVE schemes.Numerical experiments are provided to validate our theoretical results.

Superconvergence of Direct Discontinuous Galerkin Methods:Eigen-structure Analysis Based on Fourier Approach

This paper investigates superconvergence properties of the direct discontinuous Galerkin(DDG)method with interface corrections and the symmetric DDG method for diffusion equations.We apply the Fourier analysis technique to symbolically compute eigenvalues and eigenvectors of the amplification matrices for both DDG methods with different coefficient settings in the numerical fluxes.Based on the eigen-structure analysis,we carry out error estimates of the DDG solutions,which can be decomposed into three parts:(i)dissipation errors of the physically relevant eigenvalue,which grow linearly with the time and are of order 2k for P^(k)(k=2,3)approximations;(ii)projection error from a special projection of the exact solution,which is decreasing over the time and is related to the eigenvector corresponding to the physically relevant eigenvalue;(iii)dissipative errors of non-physically relevant eigenvalues,which decay exponentially with respect to the spatial mesh sizeΔx.We observe that the errors are sensitive to the choice of the numerical flux coefficient for even degree P^(2)approximations,but are not for odd degree P^(3)approximations.Numerical experiments are provided to verify the theoretical results.

Low order nonconforming mixed finite element method for nonstationary incompressible Navier-Stokes equations

This paper studies a low order mixed finite element method （FEM） for nonstationary incompressible Navier-Stokes equations. The velocity and pressure are approximated by the nonconforming constrained Q1^4ot element and the piecewise constant, respectively. The superconvergent error estimates of the velocity in the broken H^1-norm and the pressure in the L^2-norm are obtained respectively when the exact solutions are reasonably smooth. A numerical experiment is carried out to confirm the theoretical results.

A new streamline diffusion finite element method for the generalized Oseen problem

This paper aims to present a new streamline diffusion method with low order rectangular Bernardi-Raugel elements to solve the generalized Oseen equations. With the help of the Bramble-Hilbert lemma, the optimal errors of the velocity and pressure are estimated, which are independent of the considered parameter e. With an interpolation postprocessing approach, the superconvergent error of the pressure is obtained. Finally, a numerical experiment is carried out to confirm the theoretical results.

A Comparative Study of Stress Recovery Method and Error Estimation of Plate Bending Problem Using DKMQ Element

Recovery by Equilibrium in Patches (REP) is a recovery method introduced by B. Boroomand. This method is using patch as recovery media as is used by Superconvergent Patch Recovery (SPR) which is well known as a good recovery method. In this research, a numerical study of REP implementation is held to estimate error in finite element analysis using DKMQ element. The numerical study is performed with both uniform and adaptive h-type mesh refinement. The result is compared with three other recovery method, i.e. SPR method, averaging method, and projection method.

Meso-Cenozoic Evolution of Earth Surface System under the East Asian Tectonic Superconvergence

The East Asian geological setting has a long duration related to the superconvergence of the Paleo-Asian, Tethyan and Paleo-Pacific tectonic domains. The Triassic Indosinian Movement contributed to an unified passive continental margin in East Asia. The later ophiolites and I-type granites associated with subduction of the Paleo-Pacific Plate in the Late Triassic, suggest a transition from passive to active continental margins. With the presence of the ongoing westward migration of the Paleo-Pacific Subduction Zone, the sinistral transpressional stress field could play an important role in the intraplate deformation in East Asia during the Late Triassic to Middle Jurassic, being characterized by the transition from the E-W-trending structural system controlled by the Tethys and Paleo-Asian oceans to the NE-trending structural system caused by the Paleo-Pacific Ocean subduction. The continuously westward migration of the subduction zones resulted in the transpressional stress field in East Asia marked by the emergence of the Eastern North China Plateau and the formation of the Andean-type active continental margin from late Late Jurassic to Early Cretaceous （160-135 Ma）, accompanied by the development of a small amount of adakites. In the Late Cretaceous （135-90 Ma）, due to the eastward retreat of the Paleo-Pacific Subduction Zone, the regional stress field was replaced from sinistral transpression to transtension. Since a large amount of late-stage adakites and metamorphic core complexes developed, the Andean-type active continental margin was destroyed and the Eastern North China Plateau started to collapse. In the Late Cretaceous, the extension in East Asia gradually decreased the eastward retreat of the Paleo-Pacific subduction zones. Futhermore, a significant topographic inversion had taken place during the Cenozoic that resulted from a rapid uplift of the Tibet Plateau resulting from the India-Eurasian collision and the formation of the Bohai Bay Basin and other basins in the East Asian continental margin. The inversion caused a remarkable eastward migration of deformation, basin formation and magmatism. Meanwhile, the basins that mainly developed in the Paleogene resulted in a three-step topography which typically appears to drop eastward in altitude. In the Neogene, the basins underwent a rapid subsidence in some depressions after basin-controlled faulting, as well as the intracontinental extensional events in East Asia, and are likely to be a contribution to the uplift of the Tibetan Plateau.

NON-STANDARD GALERKIN METHODS OF HIGH ACCURACY FOR PARABOLIC PROBLEMS

In this paper we employ the Petrov Galerkin method for the parabolic problems to get the finite element approximate solution of high accuracy by means of the interpolation postprocessing, extrapolation and defect correction techniques.

A COUPLED CONTINUOUS-DISCONTINUOUS FEM APPROACH FOR CONVECTION DIFFUSION EQUATIONS

In this article, we introduce a coupled approach of local discontinuous Calerkin and standard finite element method for solving convection diffusion problems. The whole domain is divided into two disjoint subdomains. The discontinuous Galerkin method is adopted in the subdomain where the solution varies rapidly, while the standard finite element method is used in the other subdomain due to its lower computational cost. The stability and a priori error estimate are established. We prove that the coupled method has O（ε1/2 ＋ h1/2）hk） convergence rate in an associated norm, where ε is the diffusion coefficient, h is the mesh size and k is the degree of polynomial. The numerical results verify our theoretical results. Moreover, 2k-order superconvergence of the numerical traces at the nodes, and the optimal convergence of the errors under L2 norm are observed numerically on the uniform mesh. The numerical results also indicate that the coupled method has the same convergence order and almost the same errors as the purely LDG method.

Highly efficient H^1-Galerkin mixed finite element method (MFEM) for parabolic integro-differential equation

A highly efficient H1-Galerkin mixed finite element method （MFEM） is presented with linear triangular element for the parabolic integro-differential equation. Firstly, some new results about the integral estimation and asymptotic expansions are studied. Then, the superconvergence of order O（h^2） for both the original variable u in H1 （Ω） norm and the flux p = u in H（div, Ω） norm is derived through the interpolation post processing technique. Furthermore, with the help of the asymptotic expansions and a suitable auxiliary problem, the extrapolation solutions with accuracy O（h^3） are obtained for the above two variables. Finally, some numerical results are provided to confirm validity of the theoretical analysis and excellent performance of the proposed method.

APPLICATION OF SUPERCONVERGENCE TO A MODEL FOR COMPRESSIBLE MISCIBLE DISPLACEMENT

A modification of a finite element method of Douglas and Roberts for approximating the solution of the equations describing compressible miscible displacement in a porous medium is proposed and analyzed. The pressure is treated by a parabolic mixed finite element method using a Raviart-Thomas space of index rover a quasiregular partition, An extension of the Darcy velocity along Gauss lines is used in the evaluation of the coefficients in the Galerkin procedure for the concentration. A simple computational procedure allows the superconvergence property of the fluid velocity to be retained in our total algorithm.

Condensed Galerkin element of degree m for first-order initial-value problem with O(h^(2m+2))super-convergent nodal solutions

A new type of Galerkin finite element for first-order initial-value problems(IVPs)is proposed.Both the trial and test functions employ the same m-degreed polynomials.The adjoint equation is used to eliminate one degree of freedom(DOF)from the test function,and then the so-called condensed test function and its consequent condensed Galerkin element are constructed.It is mathematically proved and numerically verified that the condensed element produces the super-convergent nodal solutions of O(h^(2m+2)),which is equivalent to the order of accuracy by the conventional element of degree m+1.Some related properties are addressed,and typical numerical examples of both linear and nonlinear IVPs of both a single equation and a system of equations are presented to show the validity and effectiveness of the proposed element.

Superconvergence analysis of Wilson element on anisotropic meshes

The Wilson finite element method is considered to solve a class of two- dimensional second order elliptic boundary value problems. By using of the particular structure of the element and some new techniques, we obtain the superclose and global superconvergence on anisotropic meshes. Numerical example is also given to confirm our theoretical analysis.

SUPERCONVERGENCE OF THE FULL-DISCRETE F.E.M. FOR COMPRESSIBLE MISCIBLE DISPLACEMENT: THE FULL TENSOR CASE

An efficient time stepping procedure is proposed to treat the system describing compressible miscible displacement in a porous medium by employing a mixed finite element method to approximate the pressure and the fluid velocity and a standard Galerkin method to approximate the concentration. An extension of the Darcy velocity along Gauss lines is used in the evaluation of the coefficients in the Galerkin procedure for the concentration. These results show that the total algorithm has the superconvergence property of the fluid velocity.

Superconvergence of nonconforming finite element penalty scheme for Stokes problem using L^2 projection method

A modified penalty scheme is discussed for solving the Stokes problem with the Crouzeix-Raviart type nonconforming linear triangular finite element. By the L^2 projection method, the superconvergence results for the velocity and pressure are obtained with a penalty parameter larger than that of the classical penalty scheme. The numerical experiments are carried out to confirm the theoretical results.

ERROR ESTIMATES FOR THE FULL-DISCRETE MIXED FEM FOR NONLINEAR HYPERBOLIC PROBLEMS

This article treats mixed finite element methods for second order nonlinear hyperbolic e- quations.A fully discrete scheme is presented and improued L2-error estimates are established.The convergence of both the function value and the flux is demonstrated.