Nonconforming Mixed Finite Element Research Articles

Nonconforming mixed finite element method for the diffuse interface model of two-phase fluid flows

In this work, we consider a nonconforming mixed finite element method (MFEM) for a parabolic system consisting of the Cahn-Hilliard equations and the Navier-Stokes equations, which describes a diffuse interface model for the flow of two incompressible and immiscible fluids. We focus on the superconvergence analysis for the corresponding first order convex splitting scheme, where the bilinear element is used to approximate the Cahn-Hilliard equation in space, and the constrained nonconforming rotated (CNR) Q1 element together with the Q0 constant element to the Navier-Stokes equations. By use of the high accuracy properties of the elements and some specific techniques, the superclose results of order O(h2+τ) for the concentration ϕ and the chemical potential ω in the H1-norm, the velocity u in the broken H1-norm and the pressure p in L2-norm are strictly proved. Here, h is the subdivision parameter and τ, the time step. It is worth to say that we derive a new high accuracy character through the integral identity technique for the chosen elements, which plays a quite significant role in getting the final error results. Furthermore, the global superconvergence results of the corresponding variables are concluded by the interpolated postprocessing technique. Numerical tests are provided to justify the performance of the theoretical analysis.

Stabilized Nonconforming Mixed Finite Element Method for Linear Elasticity on Rectangular or Cubic Meshes

Stabilized Nonconforming Mixed Finite Element Method for Linear Elasticity on Rectangular or Cubic Meshes

The a posteriori error estimates and adaptive computation of nonconforming mixed finite elements for the Stokes eigenvalue problem

In this paper, we discuss the a posteriori error estimates and adaptive algorithm of non-conforming mixed finite elements including the Crouzeix–Raviart element and the enriched Crouzeix–Raviart element for the Stokes eigenvalue problem in Rd(d=2,3). We give the a posteriori error estimators and prove their reliability and efficiency. Based on the a posteriori error estimators we built two adaptive algorithms, the direct AFEM and the shifted-inverse AFEM. Numerical experiments and theoretical analysis are consistent, which indicates that the numerical eigenvalues obtained by the above two adaptive algorithms achieve the optimal convergence order O(dof−2d) and approximate the exact ones from below.

Dual evolution mixed variational analysis of multiphase flow in the subsurface

Dual evolution mixed variational formulations of multiphase flow models in the subsurface are established and analyzed. Specifically, compositional multiphase flow of a triphasic water–oil–gas multicomponent flow is treated, with a dual Gibbs oil–gas mass transfer model. Qualitative analysis of the mixed variational models is performed applying composition duality principles and methods. For multiphysics as well as parallel computing, macro-hybrid variational reformulations are considered. Dual semi-implicit time-marching schemes are presented as semi-discrete approximations for the flow-transport variational models, as well as for a precondition augmented exactly penalized dual Gibbs optimality variational formulation, which are all implementable as parallel proximal penalty–duality algorithms. In this manner, bases for fully discrete non-conforming macro-hybrid mixed finite element numerical realizations are rendered.

Superconvergence analysis of a nonconforming MFEM for nonlinear Schrödinger equation

In this paper, an efficient nonconforming mixed finite element method (MFEM) is studied with element and zero-order Raviart–Thomas element for a generalized nonlinear Schrödinger equation. On the one hand, by introducing as an intermediate variable, we split the equation into two low-order equations and present a semi-discrete scheme for nonconforming MFEM and prove its existence and uniqueness. And the superconvergence results for original variable u in broken -norm and flux in norm are derived by using the proved properties of the above nonconforming MFEs. On the other hand, a linearized Crank–Nicolson fully discrete scheme is constructed and the superclose and superconvergence results with order for above variables are also derived. The keys to our analysis are the following two skills: one is an important novel property for the above MFEs (see Lemma 2.3) and another is a splitting technique for nonlinear terms, while previous literature always only obtain the convergent estimates in a routine way. Finally, two numerical examples are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and τ is the time step.

Superconvergent analysis of a nonconforming mixed finite element method for time-dependent damped Navier–Stokes equations

The main aim of this paper is to develop a nonconforming mixed finite element method for the time-dependent Navier–Stokes problem with nonlinear damping term. The superconvergent analysis of a backward Euler fully-discrete scheme is presented, where the constrained nonconforming rotated (CNR) $$Q_1$$ element and the $$Q_0$$ element are used to approximate the velocity $${\varvec{u}}$$ and the pressure p, respectively. By use of the characters of the element pair together with some striking skills, i.e., mean-value skill and a new transforming skill with respect to $$\tau $$ , the superclose estimates of $$O(h^2+\tau )$$ for $${\varvec{u}}$$ in broken $$H^1$$ -norm and p in $$L^2$$ -norm are deduced rigorously. Furthermore, the global superconvergent results are obtained through the interpolated postprocessing technique. Finally, some numerical results are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and $$\tau $$ , the time step.

High Accuracy Analysis of Nonconforming Mixed Finite Element Method for the Nonlinear Sivashinsky Equation

The fourth-order nonlinear Sivashinsky equation is often used to simulate a planar solid-liquid interface for a binary alloy. In this paper, we study the high accuracy analysis of the nonconforming mixed finite element method (MFEM for short) for this equation. Firstly, by use of the special property of the nonconforming EQ1rot element (see Lemma 1), the superclose estimates of order Oh2+Δt in the broken H1-norm for the original variable u and intermediate variable p are deduced for the back-Euler (B-E for short) fully-discrete scheme. Secondly, the global superconvergence results of order Oh2+Δt for the two variables are derived through interpolation postprocessing technique. Finally, a numerical example is provided to illustrate validity and efficiency of our theoretical analysis and method.

A Framework for Nonconforming Mixed Finite Element Method for Elliptic Problems in ℝ3

In this paper, we suggest a new patch condition for nonconforming mixed finite elements (MFEs) on parallelepiped and provide a framework for the convergence. Also, we introduce a new family of nonconforming MFE space satisfying the new patch condition. The numerical experiments show that the new MFE shows optimal order convergence in Hdiv and L2-norm for various problems with discontinuous coefficient case.

NUMERICAL APPROXIMATION OF THE ELLIPTIC EIGENVALUE PROBLEM BY STABILIZED NONCONFORMING FINITE ELEMENT METHOD

<abstract> In this paper, a stabilized nonconforming mixed finite element method is used to solve the elliptic eigenvalue problem. Firstly, the lower-equal order element is used to discretize the space combined with the stabilization term based on the velocity projection method, and the error analysis is given. Moreover, the upper and lower bounds of eigenvalues are obtained. Finally, numerical experiments are carried out to verify the effectiveness of the proposed method. </abstract>

A monolithic mixed finite element method for a fluid-structure interaction problem

We propose a numerical method for modeling the interaction of a Stokes fluid and a linear elastic solid. The model problem is expressed in the stress-displacement formulation for the linear elastodynamics in the solid region and the stress-velocity formulation for the Stokes equations in the fluid region. These two systems are coupled in such a way that the interface conditions are imposed naturally in the resulting weak formulation, which is based on the Hellinger–Reissner variational principle. For the time discretization, we use a three-level scheme for each time step, with an exception at the first time step. We provide a priori error analysis for fully-discrete, nonconforming mixed finite element methods and show some numerical results to confirm our theoretical results.

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.

The families of nonconforming mixed finite elements for linear elasticity on simplex grids

We present a new family of nonconforming tetrahedral elements and a new family of nonconforming triangular elements for the stress-displacement system of linear elasticity problem. The local degrees of freedom of stress field only contain the normal moments on faces (sides) of element and the moments on element. The shape function spaces are simple, local, explicitly represented, and affine-equivalent. We also present two families simplified lowest-order finite elements by using the rigid motion model, and demonstrate our theory numerically in 2D area.

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.

Nonconforming mixed finite elements for linear elasticity on simplicial grids

This paper introduces a new family of nonconforming mixed finite elements for solving the linear elasticity equations on simplicial grids. Besides, this paper describes the construction of the lowest order basis functions. The construction only involves simple computations due to the new explicit stress shape function spaces and the procedure applies for high order cases. Numerical experiments for four benchmark problems in mechanics indicate the robust locking‐free behavior and show that the lowest order nonconforming mixed method leads to smaller stress errors than the first and second order standard Galerkin methods for the nearly incompressible case.

A new approach of superconvergence analysis of a low order nonconforming MFEM for reaction\u2013diffusion equation

In this paper, a low order nonconforming mixed finite element method (MFEM) is studied with EQ_{1}^{mathrm{rot}} element and zero order Raviart–Thomas (R–T) element for a class of nonlinear reaction–diffusion equations. On the one hand, a priori bound is proved using Lyapunov functional, which leads to the global existence and uniqueness of the approximation solutions. Further, the superclose estimates of order O(h^{2}) for original variable u in broken H^{1} norm and the flux boldsymbol{p}=nabla u in (L^{2})^{2} norm are obtained respectively for a semi-discrete scheme. On the other hand, a linearized Crank–Nicolson fully-discrete scheme is established and the superclose estimates of order O(h^{2}+tau^{2}) are also obtained unconditionally by making full use of the special characters of the above mixed finite elements (MFEs) and two different splitting techniques, which are used to deal with the consistency errors and nonlinear terms, respectively. These approaches circumvent the restrictive condition on a time step size arising as an inverse inequality used to prove a posteriori bounds in L^{infty} norm, which is necessary for nonlinear problems for the conventional finite element analysis. What is more, the corresponding global superconvergent results are derived through interpolated postprocessing techniques. Finally, numerical results are provided to confirm the theoretical analysis. Here h is the subdivision parameter and τ is the time step.

Superconvergence analysis of a new low order nonconforming MFEM for time-fractional diffusion equation

A new low order nonconforming mixed finite element method (MFEM) is proposed and analyzed for time-fractional diffusion equation with element pair (CNRQ1+Q0×Q0). A new error estimate for the consistency error of nonconforming element CNRQ1 is proved, which leads to the superclose and superconvergence results of the original variable in broken H1 norm, and of the flux in L2 norm for a fully-discrete scheme with the Caputo derivative approximated by the classical L1 method. The results obtained herein improve the corresponding conclusions in the previous literature. Finally, some numerical results are provided to confirm the theoretical analysis.

Nonconforming Finite Element Methods for Wave Propagation in Metamaterials

AbstractIn this paper, nonconforming mixed finite element method is proposed to simulate the wave propagation in metamaterials. The error estimate of the semi-discrete scheme is given by convergence order O(h2), which is less than 40 percent of the computational costs comparing with the same effect by using Nédélec-Raviart element. A Crank-Nicolson full discrete scheme is also presented with O(τ2 + h2) by traditional discrete formula without using penalty method. Numerical examples of 2D TE, TM cases and a famous re-focusing phenomena are shown to verify our theories.

Nonconforming Mixed Finite Element Method for Time-dependent Maxwell's Equations with ABC

AbstractIn this paper, a nonconforming mixed finite element method (FEM) is presented to approximate time-dependent Maxwell's equations in a three-dimensional bounded domain with absorbing boundary conditions (ABC). By employing traditional variational formula, instead of adding penalty terms, we show that the discrete scheme is robust. Meanwhile, with the help of the element's typical properties and derivative transfer skills, the convergence analysis and error estimates for semidiscrete and backward Euler fully-discrete schemes are given, respectively. Numerical tests show the validity of the proposed method.

Robust Arbitrary Order Mixed Finite Element Methods for the Incompressible Stokes Equations with pressure independent velocity errors

Standard mixed finite element methods for the incompressible Navier–Stokes equations that relax the divergence constraint are not robust against large irrotational forces in the momentum balance and the velocity error depends on the continuous pressure. This robustness issue can be completely cured by using divergence-free mixed finite elements which deliver pressure-independent velocity error estimates. However, the construction of H1 -conforming, divergence-free mixed finite element methods is rather difficult. Instead, we present a novel approach for the construction of arbitrary order mixed finite element methods which deliver pressure-independent velocity errors. The approach does not change the trial functions but replaces discretely divergence-free test functions in some operators of the weak formulation by divergence-free ones. This modification is applied to inf-sup stable conforming and nonconforming mixed finite element methods of arbitrary order in two and three dimensions. Optimal estimates for the incompressible Stokes equations are proved for the H1 and L2 errors of the velocity and the L2 error of the pressure. Moreover, both velocity errors are pressure-independent, demonstrating the improved robustness. Several numerical examples illustrate the results.

Two Mixed Finite Element Methods for Time-Fractional Diffusion Equations

Based on spatial conforming and nonconforming mixed finite element methods combined with classical L1 time stepping method, two fully-discrete approximate schemes with unconditional stability are first established for the time-fractional diffusion equation with Caputo derivative of order $$0<\alpha <1$$0<ź<1. As to the conforming scheme, the spatial global superconvergence and temporal convergence order of $$O(h^2+\tau ^{2-\alpha })$$O(h2+ź2-ź) for both the original variable u in $$H^1$$H1-norm and the flux $$\vec {p}=\nabla u$$pź=źu in $$L^2$$L2-norm are derived by virtue of properties of bilinear element and interpolation postprocessing operator, where h and $$\tau $$ź are the step sizes in space and time, respectively. At the same time, the optimal convergence rates in time and space for the nonconforming scheme are also investigated by some special characters of $$\textit{EQ}_1^{\textit{rot}}$$EQ1rot nonconforming element, which manifests that convergence orders of $$O(h+\tau ^{2-\alpha })$$O(h+ź2-ź) and $$O(h^2+\tau ^{2-\alpha })$$O(h2+ź2-ź) for the original variable u in broken $$H^1$$H1-norm and $$L^2$$L2-norm, respectively, and approximation for the flux $$\vec {p}$$pź converging with order $$O(h+\tau ^{2-\alpha })$$O(h+ź2-ź) in $$L^2$$L2-norm. Numerical examples are provided to demonstrate the theoretical analysis.