Nonconforming Finite Element Approximation Research Articles

A posteriori error estimates for nonconforming streamline-diffusion finite element methods for convection-diffusion problems

We consider residual-based a posteriori error estimation for lowest-order nonconforming finite element approximations of streamline-diffusion type for solving convection-diffusion problems. The resulting error estimator is semi-robust in the sense that it yields lower and upper bounds of the error which differ by a factor equal at most to the square root of the Peclet number. The error analysis is also shown to be applied to nonconforming finite element methods with face penalty and subgrid viscosity. Numerical results show that the estimator can be used to construct adaptive meshes.

A posterior error analysis for the nonconforming discretization of Stokes eigenvalue problem

In this paper, we present a posteriori error estimator for the nonconforming finite element approximation, including using Crouzeix-Raviart element and extended Crouzeix-Raviart element, of the Stokes eigenvalue problem. With the technique of Helmholtz decomposition, we first give out a posteriori error estimator and prove it as the global upper bound and local lower bound of the approximation error. Then, by deleting a jump term in the indicator, another simpler but equivalent indicator is obtained. Some numerical experiments are provided to verify our analysis.

Nonconforming finite element approximation of time‐dependent Maxwell's equations in Debye medium

AbstractIn this article, a new nonconforming mixed finite element method (FEM) for approximating the Maxwell's equations with Debye medium in three‐dimension are developed. By employing traditional variational formula, without adding “stability” or “penalty” terms, we show that the discrete scheme is robust. With the help of the element's typical properties, interpolation and derivative transfer skills, the convergence analysis is presented and error estimates for semidiscrete and leap‐frog fully‐discrete schemes are obtained, respectively. Numerical example shows the validity of the proposed method. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1654–1673, 2014

A posteriori error estimation based on conservative flux reconstruction for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem on anisotropic meshes

Based on conservative flux reconstruction, we derive a posteriori error estimates for the nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem on anisotropic meshes, since the solution exhibits boundary or interior layers and this anisotropy is reflected in a discretization by using anisotropic meshes. Without the assumption that the meshes are shape-regular, our estimates give the upper bounds on the error only containing the alignment measure factor, and therefore, could provide actual numerical bounds if the alignment measure was approximated very well. Two resulting error estimators are presented to be equivalent to each other up to a data oscillation, one of which can be directly constructed without solving local Neumann problems and provide computable error bound. Numerical experiments confirm that our estimates are reliable and efficient as long as the singularly perturbed problem is discretized by a suitable mesh which leads to a small alignment measure.

NONCONFORMING TETRAHEDRAL MIXED FINITE ELEMENTS FOR ELASTICITY

This paper presents a nonconforming finite element approximation of the space of symmetric tensors with square integrable divergence, on tetrahedral meshes. Used for stress approximation together with the full space of piecewise linear vector fields for displacement, this gives a stable mixed finite element method which is shown to be linearly convergent for both the stress and displacement, and which is significantly simpler than any stable conforming mixed finite element method. The method may be viewed as the three-dimensional analogue of a previously developed element in two dimensions. As in that case, a variant of the method is proposed as well, in which the displacement approximation is reduced to piecewise rigid motions and the stress space is reduced accordingly, but the linear convergence is retained.

Star-Based a Posteriori Error Estimates for Elliptic Problems

We give an a posteriori error estimator for low order nonconforming finite element approximations of diffusion-reaction and Stokes problems, which relies on the solution of local problems on stars. It is proved to be equivalent to the energy error up to a data oscillation, without requiring Helmholtz decomposition of the error nor saturation assumption. Numerical experiments illustrate the good behavior and efficiency of this estimator for generic elliptic problems.

A posterior error estimator and lower bound of a nonconforming finite element method

In this paper, we present an a posteriori error estimator and the lower bound for a nonconforming finite element approximation, i.e. the extended Crouzeix–Raviart element, of the Laplace eigenvalue problem. Under the guideline of the analysis to the Laplace source problem, we first give out an error indicator and prove it as the global upper and local lower bounds of the approximation error. We also give the lower-bound analysis for this type of nonconforming element on the adaptive meshes. Some numerical experiments are presented to verify our theoretical results.

A posteriori error estimates for non conforming approximation of quasi Stokes problem

We derive and analyze an a posteriori error estimator for nonconforming finite element approximation for the quasi-Stokes problem, which is based on the solution of local problem on stars with low cost computation, this indicator is equivalent to the energy error norm up to data oscillation, neither saturation assumption nor comparison with residual estimator are made.

Superconvergence analysis and extrapolation of quasi-Wilson nonconforming finite element method for nonlinear Sobolev equations

Quasi-Wilson nonconforming finite element approximation for a class of nonlinear Sobolev equations is discussed on rectangular meshes. We first prove that this element has two special characters by novel approaches. One is that (∇h(u − Ihu),∇hvh)h may be estimated as order O(h2) when u ∈ H3(Ω), where Ihu denotes the bilinear interpolation of u, vh is a polynomial belongs to quasi-Wilson finite element space and ∇h denotes the piecewise defined gradient operator, h is the mesh size tending to zero. The other is that the consistency error of this element is of order O(h2)/O(h3) in broken H1-norm, which is one/two order higher than its interpolation error when u ∈ H3(Ω)/H4(Ω). Then we derive the optimal order error estimate and superclose property via mean-value method and the known high accuracy result of bilinear element. Furthermore, we deduce the global superconvergence through interpolation post processing technique. At last, an extrapolation result of order O(h3), two order higher than traditional error estimate, is obtained by constructing a new suitable extrapolation scheme.

Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations

The paper deals with error estimates and lower bound approximations of the Steklov eigenvalue problems on convex or concave domains by nonconforming finite element methods. We consider four types of nonconforming finite elements: Crouzeix-Raviart, Q 1 rot , EQ 1 rot and enriched Crouzeix-Raviart. We first derive error estimates for the nonconforming finite element approximations of the Steklov eigenvalue problem and then give the analysis of lower bound approximations. Some numerical results are presented to validate our theoretical results.

EQ 1 rot nonconforming finite element approximation to Signorini problem

In this paper, we apply EQ 1 rot nonconforming finite element to approximate Signorini problem. If the exact solution u ∈ H 5/2 (Ω), the error estimate of order O ( h ) about the broken energy norm is obtained for quadrilateral meshes satisfying regularity assumption and bi-section condition. Furthermore, the superconvergence results of order O ( h 3/2 ) are derived for rectangular meshes. Numerical results are presented to confirm the considered theory.

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.

An Anisotropic Locking-Free Nonconforming Triangular Finite Element Method for Planar Linear Elasticity Problem

The main aim of this paper is to study the nonconforming linear triangular CrouzeixRaviart type finite element approximation of planar linear elasticity problem with the pure displacement boundary value on anisotropic general triangular meshes satisfying the maximal angle condition and coordinate system condition. The optimal order error estimates

Analysis of a projection method for low-order nonconforming finite elements

We present a study of the incremental projection method to solve incompressible unsteady Stokes equations based on a low degree non-conforming finite element approximation in space, with, in particular, a piecewise constant approximation for the pressure. The numerical method falls in the class of algebraic projection methods. We provide an error analysis in the case of Dirichlet boundary conditions, which confirms that the splitting error is second-order in time. In addition, we show that pressure artificial boundary conditions are present in the discrete pressure elliptic operator, even if this operator is obtained by a splitting performed at the discrete level; however, these boundary conditions are imposed in the finite volume (weak) sense and the optimal order of approximation in space is still achieved, even for open boundary conditions.

Convergence and superconvergence analysis of a nonconforming finite element method for solving the Signorini problem

In this paper, we present the Carey nonconforming finite element approximation of the variational inequality resulting from the Signorini problem. Firstly, we show that if the displacement field is of H2-regularity, the optimal convergence rate of O(h) can be obtained with respect to the energy norm. Secondly, if stronger but reasonable H52-regularity is available, the superconvergence rate of O(h32) can be derived through the interpolated postprocessing technique. Finally, numerical experiments are given which are consistent with our theoretical analysis.

二维非定常Navier-Stokes方程的Euler隐/显格式子格涡旋粘性非协调有限元法

本文研究了二维高雷诺数情形下，非定常不可压Navier-Stokes方程的Euler隐/显格式子格涡旋粘性非协调有限元方法。隐式处理线性项，避免了时间步长的苛刻限制，显式处理非线性项，使得对所有时间层求解时，系统矩阵为同一常数矩阵；时间项做向后Euler差分离散，空间用C-R非协调有限元逼近，构造子格涡旋粘性有限元方法，克服了在情形下Galerkin有限元方法的不稳定现象。本文改善了稳定性下对时间步长的限制，并给出了不依赖粘性系数的速度和压力误差估计。 In this paper, an Euler implicit/explicit scheme with nonconforming finite element method of subgrid eddy viscosity type for solving the 2D nonstationary incompressible Navier-Stokes equations under high Reynolds number is considered. The implicit/explicit scheme which is implicit for the linear terms and explicit for the nonlinear term, avoids the severely restricted time step size from stability requirement and results in a linear system with a same constant matrix at each level of time. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation is used for the velocity and pressure field with the subgrid eddy viscosity technique, to cope with usual instabilities caused by Galerkin finite element methods. This paper also improved the restricted time step size which under stable conditions and given error estimates of velocity and pressure which independent on the viscosity .

Computable error bounds for nonconforming Fortin-Soulie finite element approximation of the Stokes problem

We propose computable a posteriori error estimates for a second order nonconforming finite element approximation of the Stokes problem. The estimator is completely free of unknown constants and gives a guaranteed numerical upper bound on the error, in terms of a lower bound for the inf-sup constant of the underlying continuous problem. The estimator is also shown to provide a lower bound on the error up to a constant and higher order data oscillation terms. Numerical results are presented illustrating the theory and the performance of the estimator.

An L2‐stable approximation of the Navier–Stokes convection operator for low‐order non‐conforming finite elements

AbstractWe develop in this paper a discretization for the convection term in variable density unstationary Navier–Stokes equations, which applies to low‐order non‐conforming finite element approximations (the so‐called Crouzeix–Raviart or Rannacher–Turek elements). This discretization is built by a finite volume technique based on a dual mesh. It is shown to enjoy an L2 stability property, which may be seen as a discrete counterpart of the kinetic energy conservation identity. In addition, numerical experiments confirm the robustness and the accuracy of this approximation; in particular, in L2 norm, second‐order space convergence for the velocity and first‐order space convergence for the pressure are observed. Copyright © 2010 John Wiley & Sons, Ltd.

Nonconforming finite element approximation of the Giesekus model for polymer flows

We present a numerical approximation of the Giesekus equation which is considered as a realistic model for polymer flows. We use nonconforming finite elements on quadrilateral grids which necessitate the addition of two stabilization terms. An appropriate upwind scheme is employed for the convective term. The underlying discrete Stokes problem is then analysed. Finally, numerical tests are presented in order to validate the code, illustrating its good behavior for large Weissenberg numbers. Comparisons with Polyflow® and with the literature are also carried out.

A streamline diffusion nonconforming finite element method for the time-dependent linearized Navier-Stokes equations

A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming (P1) 2 − P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.