Polynomial Preserving Recovery Research Articles

Polynomial preserving recovery for the finite volume element methods under simplex meshes

The recovered gradient, using the polynomial preserving recovery (PPR), is constructed for the finite volume element method (FVEM) under simplex meshes. Regarding the main results of this paper, there are two aspects. Firstly, we investigate the supercloseness property of the FVEM, specifically examining the quadratic FVEM under tetrahedral meshes. Secondly, we present several guidelines for selecting computing nodes such that the least-squares fitting procedure of the PPR admits a unique solution. Numerical experiments demonstrate that the recovered gradient by the PPR exhibits superconvergence.

An immersed selective discontinuous Galerkin method in particle-in-cell simulation with adaptive Cartesian mesh and polynomial preserving recovery

In this paper, a selective discontinuous Galerkin (SDG) method and a polynomial preserving recovery (PPR) method are developed and integrated with the immersed-finite-element particle-in-cell (IFE-PIC) method, in order to carry out the plasma-material interaction simulation on adaptive Cartesian meshes (ACM). To significantly save the computational cost in practice, the PIC simulation often needs to use Cartesian meshes for the whole domain and the ACM for some focused regions in the plasma-material interaction problems. Therefore, we utilize the SDG method with immersed finite elements for the field solver of the IFE-PIC method, as the key technique to allow the local Cartesian mesh refinement which will generate hanging nodes. To minimize the degree of freedom of the SDG on the ACM and reduce the computational cost, a new selective technique of the discontinuous global basis functions is proposed for the SDG. Various types of nodes in the ACM are discussed for the implementation of this selective technique, including the hanging nodes. Meanwhile, the gathering and scattering steps of the PIC simulation also need to be appropriately performed on the ACM. The PPR method is a generic gradient recovery technique to be incorporated into the IFE-PIC method for more accurate force deposition on the ACM. In addition, two other algorithmic structures of the traditional IFE-PIC method have been modified to accommodate the ACM in the simulation, including a new mapping array structure for the particle positioning and a new charge deposition with some special particle positions in the ACM. As a result, the integrated immersed-selective-discontinuous-Galerkin particle-in-cell (ISDG-PIC) method can efficiently perform the plasma-material interactions simulation on the Cartesian meshes with local mesh refinement independent of the interface. Several numerical experiments are performed to demonstrate the convergence, efficiency, and applicability of the proposed ISDG-PIC method.

The superconvergence gradient recovery method for linear finite element method with polygons

AbstractTo identify the superconvergent areas of gradient solutions is a worthwhile part of gradient recovery in finite element method. Compared with triangular and quadrilateral elements, the position determination for the superconvergent gradient of polygonal element is much more difficult because of the non‐polynomial form of interpolation functions. In this article, the gradient superconvergence points in C0 polygonal element is discovered, and its existence is proved theoretically. By dividing the canonical polygonal element into sub‐triangles, a set of provisional triangular elements is generated to replace the existing polygon element. The superconvergent results of gradient solutions are obtained at the centroids of sub‐triangles. Based on that, the gradient recovery method using superconvergent patch recovery (SPR) is developed for general polygonal mesh. Numerical examples for the Poisson equation are solved and the simulated results compare with polynomial preserving recovery (PPR) to demonstrate the accuracy and convergence properties of the proposed method. The extend application of the superconvergence recovery method for adaptive analysis is also presented at the end.

Superconvergence analysis of linear FEM based on polynomial preserving recovery for Helmholtz equation with high wave number

Although superconvergence properties of the polynomial preserving recovery (PPR) for finite element methods (FEM) for coercive elliptic problems are well-understood, the superconvergence behavior of PPR for the Helmholtz equation with high wave number still remains unclear. We study the superconvergence property of the linear FEM with PPR for the two dimensional Helmholtz equation on triangulations satisfying the O(h1+α) approximate parallelogram property for some constant α>0. We prove the supercloseness on the H1-seminorm between the finite element solution and the linear interpolant of the exact solution under k(kh)2≤C0, where k is the wave number and h is the mesh size. Then, we obtain the superconvergence result based on the PPR technique which says that the PPR improves only the interpolation error but keeps the pollution error unchanged. Finally, we provide numerical tests to verify the superconvergence property and to demonstrate that the PPR combining with the continuous interior penalty technique is much effective for improving both the interpolation error and the pollution error.

Superconvergence Analysis of the Polynomial Preserving Recovery for Elliptic Problems with Robin Boundary Conditions

Superconvergence Analysis of the Polynomial Preserving Recovery for Elliptic Problems with Robin Boundary Conditions

Parametric Polynomial Preserving Recovery on Manifolds

This paper investigates gradient recovery schemes for data defined on discretized manifolds. The proposed method, parametric polynomial preserving recovery (PPPR), does not require the tangent spac...

Supercloseness of Linear DG-FEM and Its Superconvergence Based on the Polynomial Preserving Recovery for Helmholtz Equation

In this paper we study the supercloseness property of the linear discontinuous Galerkin (DG) finite element method and its superconvergence behavior after post-processing by the polynomial preserving recovery (PPR). The error estimate with explicit dependence on the wave number k, the penalty parameter $$\mu $$ and the mesh condition parameter $$\alpha $$ is derived. We prove the supercloseness between the DG finite element solution and the linear interpolation and the superconvergence for the recovered gradient by the PPR under the assumption $$k(kh)^2\le C_0$$ (h is the mesh size) and certain mesh conditions. Furthermore, we estimate the error between the DG numerical gradient and recovered gradient, which motivates us to define the a posteriori error estimator and design a Richardson extrapolation to post-process the recovered gradient by PPR. Finally, some numerical examples are provided to confirm the theoretical results of superconvergence analysis.

Supercloseness analysis and polynomial preserving Recovery for a class of weak Galerkin Methods

In this article, we analyze convergence and supercloseness properties of a class of weak Galerkin (WG) finite element methods for solving second‐order elliptic problems. It is shown that the WG solution is superclose to the Lagrange interpolant using Lobatto points. This supercloseness behavior is obtained through some newly designed stabilization terms. A postprocessing technique using polynomial preserving recovery (PPR) is introduced for the WG approximation. Superconvergence analysis is performed for the PPR recovered gradient. Numerical examples are provided to illustrate our theoretical results.

Superconvergence Analysis of Gradient Recovery Method for TM Model of Electromagnetic Scattering in the Cavity

AbstractIn this paper, we consider the transform magnetic (TM) model of electromagnetic scattering in the cavity. By the Polynomial Preserving Recovery technique, we present superconvergence analysis for the vertex-edge-face type finite element. From the numerical example, we can see that the provided method is efficient and stable.

Superconvergence Analysis of Gradient Recovery Method for TM Model of Electromagnetic Scattering in the Cavity

AbstractIn this paper, we consider the transform magnetic (TM) model of electromagnetic scattering in the cavity. By the Polynomial Preserving Recovery technique, we present superconvergence analysis for the vertex-edge-face type finite element. From the numerical example, we can see that the provided method is efficient and stable.

Finite Element Model for Linear Elastic Thick Shells Using Gradient Recovery Method

This research purposed a new family of finite elements for spherical thick shell based on Nzengwa-Tagne’s model proposed in 1999. The model referred to hereafter as N-T model contains the classical Kirchhoff-Love (K-L) kinematic with additional terms related to the third fundamental form governing strain energy. Transverse shear stresses are computed andC0finite element is proposed for numerical implementation. However, using straight line triangular elements does not guarantee a correct computation of stress across common edges of adjacent elements because of gradient jumps. The gradient recovery method known as Polynomial Preserving Recovery (PPR) is used for local interpolation and applied on a hemisphere under diametrically opposite charges. A good agreement of convergence results is observed; numerical results are compared to other results obtained with the classical K-L thin shell theory. Moreover, simulation on increasing values of the ratio of the shell shows impact of the N-T model especially on transverse stresses because of the significant energy contribution due to the third fundamental form tensor present in the kinematics of this model. The analysis of the thickness ratio shows difference between the classical K-L theory and N-T model when the ratio is greater than 0.099.

Polynomial Preserving Recovery for High Frequency Wave Propagation

Polynomial preserving recovery (PPR) was first proposed and analyzed in Zhang and Naga in SIAM J Sci Comput 26(4):1192---1213, (2005), with intensive following applications on elliptic problems. In this paper, we generalize the study of PPR to high-frequency wave propagation. Specifically, we establish the supercloseness between finite element solution and its interpolation with explicit dependence on the frequency of wavefield, and then prove the superconvergence of PPR for high-frequency solutions to wave equation based on the supercloseness. We also present several numerical examples of PPR for both low-frequency and high-frequency wave propagation in order to confirm the theoretical results of superconvergence analysis.

A Type of Finite Element Gradient Recovery Method based on Vertex-Edge-Face Interpolation: The Recovery Technique and Superconvergence Property

AbstractIn this paper, a new type of gradient recovery method based on vertex-edge-face interpolation is introduced and analyzed. This method gives a new way to recover gradient approximations and has the same simplicity, efficiency, and superconvergence properties as those of superconvergence patch recovery method and polynomial preserving recovery method. Here, we introduce the recovery technique and analyze its superconvergence properties. We also show a simple application in the a posteriori error estimates. Some numerical examples illustrate the effectiveness of this recovery method.

Can We Have Superconvergent Gradient Recovery Under Adaptive Meshes?

We study adaptive finite element methods for elliptic problems with domain corner singularities. Our model problem is the two-dimensional Poisson equation. Results of this paper are twofold. First, we prove that there exists an adaptive mesh (gauged by a discrete mesh density function) under which the recovered gradient by the polynomial preserving recovery (PPR) is superconvergent. Second, we demonstrate by numerical examples that an adaptive procedure with an a posteriori error estimator based on PPR does produce adaptive meshes that asatisfy our mesh density assumption, and the recovered gradient by PPR is indeed superconvergent in the adaptive process.

Enhancing Eigenvalue Approximation by Gradient Recovery

The polynomial preserving recovery (PPR) is used to enhance the finite element eigenvalue approximation. Remarkable fourth order convergence is observed for linear elements under structured meshes as well as unstructured initial meshes (produced by the Delaunay triangulation) with the conventional bisection refinement.

The polynomial-preserving recovery for higher order finite element methods in 2D and 3D

The Polynomial-Preserving Recovery (PPR) technique is extended to recover continuous gradients from $C^0$ finite element solutions of an arbitrary order in 2D and 3D problems. The stability of the PPR is theoretically investigated in a general framework. In 2D, the stability is established under a simple geometric condition. The numerical experiments demonstrated that the PPR-recovered gradient enjoys superconvergence, and the Zienkiewicz-Zhu error estimator based on the PPR-recovered gradient is asymptotically exact.

Validation of thea posteriorierror estimator based on polynomial preserving recovery for linear elements

AbstractThe purpose of this work is to investigate the quality of thea posteriorierror estimator based on the polynomial preserving recovery (PPR). The main tool in this investigation is the computer‐based theory. Also, a comparison is made between this estimator and the one based on the superconvergence patch recovery (SPR). The results of this comparison were found to be in favour of the estimator based on the PPR. Copyright © 2004 John Wiley & Sons, Ltd.

A Posteriori Error Estimates Based on the Polynomial Preserving Recovery

Superconvergence of order $O(h^{1+\rho})$, for some $\rho > 0$, is established for the gradient recovered with the polynomial preserving recovery (PPR) when the mesh is mildly structured. Consequently, the PPR-recovered gradient can be used in building an asymptotically exact a posteriori error estimator.