Low Order Nonconforming Finite Element Research Articles

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

Low order nonconforming finite element methods for nuclear reactor model

A uniform framework of the linearized fully decoupled fully discrete schemes is developed and investigated with low order nonconforming finite element methods (FEMs) for nuclear reactor model. On the one hand, a general scheme called Scheme I is constructed. Then, the modified Ritz projection and mathematical induction are used to obtain its unconditional optimal error estimate in the L2-norm and superclose estimates in the broken H1-norm, thereby correcting the mistake in previous literature and eliminating the need for the so-called popular time-space splitting method. On the other hand, a new positivity-preserving scheme named Scheme II, which combines the traditional FEMs with cut-off post-processing method, is designed to avoid the non-physical oscillation in numerical calculations. Based on the results of Scheme I, an optimal error estimate in the L2-norm for Scheme II is also derived. Finally, taking the nonconforming EQ1rot element as an example, some numerical experiments are provided to demonstrate the theoretical results. It is worth noting that the analysis presented herein is also suitable for simulating nuclear reactor model in the narrow channel with anisotropic meshes.

Uniformly superconvergent analysis of an efficient two-grid method for nonlinear Bi-wave singular perturbation problem

The main aim of this paper is to present a two-grid method for the fourth order nonlinear Bi-wave singular perturbation problem with low order nonconforming finite element based on the Ciarlet–Raviart scheme. The existence and uniqueness of the approximation solution are demonstrated through the Brouwer fixed point theorem and the uniform superconvergent estimates in the broken H1− norm and L2− norm are obtained, which are independent of the perturbation parameter δ. Some numerical results indicate that the proposed method is indeed an efficient algorithm.

A nonconforming finite element method for the stationary Smagorinsky model

In this paper, we focus on a low order nonconforming finite element method (FEM) for the stationary Smagorinsky model. The velocity and pressure are approximated by the constrained nonconforming rotated Q1 element (CN Q1rot) and piecewise constant element, respectively. Optimal error estimates of the velocity in the broken H1-norm and L2-norm, and the pressure in the L2-norm are derived by some nonlinear analysis techniques and Aubin-Nitsche duality argument. The supercloseness and superconvergent results are also obtained under some reasonable regularity assumptions. Finally, a numerical example is implemented to confirm our theoretical analysis.

Superconvergence analysis of low order nonconforming finite element methods for variational inequality problem with displacement obstacle

Superconvergence analysis of nonconforming finite element methods (FEMs) are discussed for solving the second order variational inequality problem with displacement obstacle. The elements employed have a common typical character, i.e., the consistency error can reach order O(h3/2−ɛ), nearly 1/2 order higher than their interpolation error when the exact solution of the considered problem belongs to H5/2−ɛ(Ω) for any ε > 0. By making full use of special properties of the element’s interpolations and Bramble–Hilbert lemma, the superconvergence error estimates of order O(h3/2−ɛ) in the broken H1-norm are derived. Finally, some numerical results are provided to confirm the theoretical results.

Two low-order nonconforming finite element methods for the Stokes flow in three dimensions

Abstract In this paper, we propose two low-order nonconforming finite element methods (FEMs) for the three-dimensional Stokes flow that generalize the nonconforming FEM of Kouhia & Stenberg (1995, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Eng, 124, 195–212). The finite element spaces proposed in this paper consist of two globally continuous components (one piecewise affine and one enriched component) and one component that is continuous at the midpoints of interior faces. We prove that the discrete Korn inequality and a discrete inf–sup condition hold uniformly in the mesh size and also for a nonempty Neumann boundary. Based on these two results, we show the well-posedness of the discrete problem. Two counterexamples prove that there is no direct generalization of the Kouhia–Stenberg FEM to three space dimensions: the finite element space with one nonconforming and two conforming piecewise affine components does not satisfy a discrete inf–sup condition with piecewise constant pressure approximations, while finite element functions with two nonconforming and one conforming component do not satisfy a discrete Korn inequality.

A low order characteristic-nonconforming finite element method for nonlinear Sobolev equation with convection-dominated term

A low order nonconforming finite element method is combined with the method of characteristics to treat the nonlinear Sobolev equation with convection-dominated term. The optimal order error estimates in a broken H1-norm and L2-norm are obtained by some special properties of the interpolation operator and the mean value trick, instead of the so-called elliptic projection which is an indispensable tool in the convergence analysis of the previous literature. In addition, the global superconvergence is derived based on the interpolated postprocessing technique. The theoretical results are confirmed by some numerical experiments.

Lower Bounds for Eigenvalues of Elliptic Operators: By Nonconforming Finite Element Methods

The paper is to introduce a new systematic method that can produce lower bounds for eigenvalues. The main idea is to use nonconforming finite element methods. The conclusion is that if local approximation properties of nonconforming finite element spaces are better than total errors (sums of global approximation errors and consistency errors) of nonconforming finite element methods, corresponding methods will produce lower bounds for eigenvalues. More precisely, under three conditions on continuity and approximation properties of nonconforming finite element spaces we analyze abstract error estimates of approximate eigenvalues and eigenfunctions. Subsequently, we propose one more condition and prove that it is sufficient to guarantee nonconforming finite element methods to produce lower bounds for eigenvalues of symmetric elliptic operators. We show that this condition hold for most low-order nonconforming finite elements in literature. In addition, this condition provides a guidance to modify known nonconforming elements in literature and to propose new nonconforming elements. In fact, we enrich locally the Crouzeix-Raviart element such that the new element satisfies the condition; we also propose a new nonconforming element for second order elliptic operators and prove that it will yield lower bounds for eigenvalues. Finally, we prove the saturation condition for most nonconforming elements.

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.

Low Order Nonconforming Expanded Characteristic- Mixed Finite Element Method for the Convection- Diffusion Problem

A low order nonconforming finite element method is proposed for the convection-diffusion equations with the expanded characteristic-mixed finite element scheme. The method is a combination of characteristic approximation to handle the convection part in time and a expanded nonconforming mixed finite element spatial approximation to deal with the diffusion part. In the process, the interpolation operator is employed instead of the so-called elliptic projection which is an indispensable tool used for the convergence analysis in the previous literature. When the exact solutions belong to H 2 (Ω) instead of H 3 (Ω), the corresponding optimal order error estimates in L 2 -norm are obtained by use of some distinct properties of the nonconforming finite elements.

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.

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.

A posteriori error estimators for nonconforming finite element methods of the linear elasticity problem

In this work we derive and analyze a posteriori error estimators for low-order nonconforming finite element methods of the linear elasticity problem on both triangular and quadrilateral meshes, with hanging nodes allowed for local mesh refinement. First, it is shown that equilibrated Neumann data on interelement boundaries are simply given by the local weak residuals of the numerical solution. The first error estimator is then obtained by applying the equilibrated residual method with this set of Neumann data. From this implicit estimator we also derive two explicit error estimators, one of which is similar to the one proposed by Dörfler and Ainsworth (2005) [24] for the Stokes problem. It is established that all these error estimators are reliable and efficient in a robust way with respect to the Lamé constants. The main advantage of our error estimators is that they yield guaranteed, i.e., constant-free upper bounds for the energy-like error (up to higher order terms due to data oscillation) when a good estimate for the inf–sup constant is available, which is confirmed by some numerical results.

A low order nonconforming anisotropic finite element approximation to parabolic problem

A low order nonconforming finite element is applied to the parabolic problem with anisotropic meshes. Both the semidiscrete and fully discrete forms are studied. Some superclose properties and superconvergence are obtained through some novel approaches and techniques.