Nonconforming Finite Element Method Research Articles

Nonconforming Finite Element Methods for Two-Dimensional Linearly Elastic Shallow Shell Model

Nonconforming Finite Element Methods for Two-Dimensional Linearly Elastic Shallow Shell Model

Numerical analysis of optimal control problems governed by fourth‐order linear elliptic equations using the Hessian discretization method

AbstractThis article focuses on the optimal control problems governed by fourth‐order linear elliptic equations with clamped boundary conditions in the framework of the Hessian discretization method (HDM). The HDM is an abstract framework that enables the convergence analysis of several numerical methods such as the conforming finite element methods, the Adini and Morley nonconforming finite element methods (ncFEMs), the method based on gradient recovery (GR) operators, and the finite volume methods (FVMs). Basic error estimates and superconvergence results are established for the state, adjoint, and control variables. A companion operator for the GR method with specific property is designed. The article concludes with numerical results that illustrate the theoretical convergence rates for GR method, Adini ncFEM, and FVM.

Superconvergence analysis of the nonconforming FEM for the Allen–Cahn equation with time Caputo–Hadamard derivative

This paper first introduces a high-order approximate formula for the Caputo–Hadamard fractional derivative using interpolation theory, and analyzes its convergence and accuracy. Although the formula bears resemblance to the logarithmic L2-1σ formula in form, it differs in the selection of mesh points, making it suitable for more complex models in numerical computations. Subsequently, a numerical method for the Allen–Cahn equation with time Caputo–Hadamard derivative is proposed. The temporal direction is discretized using the newly derived difference formula, while the spatial direction is approximated by the anisotropic nonconforming quasi-Wilson finite element method (FEM). It is proven that the derived scheme has optimal convergence accuracy in the L2-norm and global superconvergence property in the H1-norm. Numerical examples are provided to corroborate the theoretical claims.

Pointwise adaptive non-conforming finite element method for the obstacle problem

Pointwise adaptive non-conforming finite element method for the obstacle problem

Adaptive finite element methods for scalar double‐well problem

AbstractSome scalar double‐well problems eventually lead to a degenerate convex minimization problem with unique stress. We consider the adaptive conforming and nonconforming finite element methods for the scalar double‐well problem with the reliable a posteriori error analysis. A number of experiments confirm the effective decay rates of the methods.

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.

Convergence of Adaptive Crouzeix–Raviart and Morley FEM for Distributed Optimal Control Problems

Abstract This article discusses the quasi-optimality of adaptive nonconforming finite element methods for distributed optimal control problems governed by 𝑚-harmonic operators for m = 1 , 2 m=1,2 . A variational discretization approach is employed and the state and adjoint variables are discretized using nonconforming finite elements. Error equivalence results at the continuous and discrete levels lead to a priori and a posteriori error estimates for the optimal control problem. The general axiomatic framework that includes stability, reduction, discrete reliability, and quasi-orthogonality establishes the quasi-optimality. Numerical results demonstrate the theoretically predicted orders of convergence and the efficiency of the adaptive estimator.

A multi-level correction scheme for biharmonic eigenvalue problem by nonconforming finite element method

A multi-level correction scheme for biharmonic eigenvalue problem by nonconforming finite element method

Surface Crouzeix‐Raviart element for the Bochner Laplacian equation

AbstractRecently, a nonconforming finite element method has been derived for vector valued flow problems on the sphere. In the approach, the flow is approximated via edge integration through the Crouzeix‐Raviart element. The discretization is realized on local flat triangles which coincide with the surface of the sphere on the edge midpoints. In this contribution, we derive an energy error estimate for this discretization that takes into account the geometric error as well as the error to the solution of the partial differential equation. The analysis is performed for a vector Laplace problem which includes covariant derivatives of tangential vector fields. The latter are closely related to operators that occur in flow problems on the surface.

Nonconforming Finite Element Methods of Order Two and Order Three for the Stokes Flow in Three Dimensions

Nonconforming Finite Element Methods of Order Two and Order Three for the Stokes Flow in Three Dimensions

A New Immersed Finite Element Method for Two-Phase Stokes Problems Having Discontinuous Pressure

Abstract In this paper, we develop a new immersed finite element method (IFEM) for two-phase incompressible Stokes flows. We allow the interface to cut the finite elements. On the noninterface element, the standard Crouzeix–Raviart element and the P 0 {P_{0}} element pair is used. On the interface element, the basis functions developed for scalar interface problems (Kwak et al., An analysis of a broken P 1 {P_{1}} -nonconforming finite element method for interface problems, SIAM J. Numer. Anal. (2010)) are modified in such a way that the coupling between the velocity and pressure variable is different. There are two kinds of basis functions. The first kind of basis satisfies the Laplace–Young condition under the assumption of the continuity of the pressure variable. In the second kind, the velocity is of bubble type and is coupled with the discontinuous pressure, still satisfying the Laplace–Young condition. We remark that in the second kind the pressure variable has two degrees of freedom on each interface element. Therefore, our methods can handle the discontinuous pressure case. Numerical results including the case of the discontinuous pressure variable are provided. We see optimal convergence orders for all examples.

Superconvergence analysis of nonconforming finite element method for two-dimensional time-fractional Allen–Cahn equation

In this paper, we investigate the numerical solution of the time-fractional Allen–Cahn equation with variable diffusion coefficients. A fully discrete approximation is presented by the L2-1σ scheme on graded meshes and spatial high precision nonconforming finite element method (FEM). The nonlinear term is treated with the Newton linearization method. Based on the modified discrete fractional Grönwall inequality, we rigorously prove that the proposed scheme can achieve optimal convergence accuracy in both time and space directions. Furthermore, the H1-norm global superconvergence result is obtained through the interpolation postprocessing technique. Finally, numerical experiments confirm the correctness of our theoretical analysis.

A novel error analysis of nonconforming finite element for the clamped Kirchhoff plate with elastic unilateral obstacle

<abstract><p>A nonconforming finite element method (FEM) is proposed and analyzed for the clamped thin elastic Kirchhoff plate unilaterally constrained by an elastic obstacle. The discrete scheme is constructed by using the strongly discontinuous Bergan's energy-orthogonal plate element, which has simple degrees of freedom and about 25 percent fewer global dimension than that of the famous triangular Morley element. A novel error analysis is presented to overcome the difficulties caused by the strong discontinuity and derive the optimal estimate. Numerical experiments are carried out to verify the theoretical analysis.</p></abstract>

Convergence and superconvergence analysis for nonlinear delay reaction–diffusion system with nonconforming finite element

AbstractIn this article, we propose and analyze several numerical methods for the nonlinear delay reaction–diffusion system with smooth and nonsmooth solutions, by using Quasi‐Wilson nonconforming finite element methods in space and finite difference methods (including uniform and nonuniform L1 and L2‐1σ schemes) in time. The optimal convergence results in the senses of L2‐norm and broken H1‐norm, and H1‐norm superclose results are derived by virtue of two types of fractional Grönwall inequalities. Then, the interpolation postprocessing technique is used to establish the superconvergence results. Moreover, to improve computational efficiency, fast algorithms by using sum‐of‐exponential technique are built for above proposed numerical schemes. Finally, we present some numerical experiments to confirm the theoretical correctness and show the effectiveness of the fast algorithms.

A new energy-stable nonconforming finite element method for Sobolev equation with Burgers’ type nonlinearity

We propose an energy-stable finite element method (FEM) of the Sobolev equation with Burgers’ type nonlinearity by implicit Euler method in time and nonconforming EQ1rot element in space. A stabilized term is innovatively added in this work to preserve the energy dissipation of the numerical scheme. Then we directly obtain the prior estimates of the numerical solution, with which the unique solvability of the fully-discrete scheme is derived. Subsequently, a novel strategy is utilized to achieve the superclose estimate of order O(h2+τ) in broken H1-norm. Using the interpolated postprocessing technique, the global superconvergence result is obtained. At last, numerical example is given to confirm the theoretical analysis. Here, h is the spatial parameter, and τ is the time step.

Robust Monolithic Solvers for the Stokes--Darcy Problem with the Darcy Equation in Primal Form

We construct mesh-independent and parameter-robust monolithic solvers for the coupled primal Stokes-Darcy problem. Three different formulations and their discretizations in terms of conforming and non-conforming finite element methods and finite volume methods are considered. In each case, robust preconditioners are derived using a unified theoretical framework. In particular, the suggested preconditioners utilize operators in fractional Sobolev spaces. Numerical experiments demonstrate the parameter-robustness of the proposed solvers.

Optimal H1-Norm Estimation of Nonconforming FEM for Time-Fractional Diffusion Equation on Anisotropic Meshes

In this paper, based on the L2-1σ scheme and nonconforming EQ1rot finite element method (FEM), a numerical approximation is presented for a class of two-dimensional time-fractional diffusion equations involving variable coefficients. A novel and detailed analysis of the equations with an initial singularity is described on anisotropic meshes. The fully discrete scheme is shown to be unconditionally stable, and optimal second-order accuracy for convergence and superconvergence can be achieved in both time and space directions. Finally, the obtained numerical results are compared with the theoretical analysis, which verifies the accuracy of the proposed method.

Optimal Convergence Analysis of Two-Level Nonconforming Finite Element Iterative Methods for 2D/3D MHD Equations.

Several two-level iterative methods based on nonconforming finite element methods are applied for solving numerically the 2D/3D stationary incompressible MHD equations under different uniqueness conditions. These two-level algorithms are motivated by applying the m iterations on a coarse grid and correction once on a fine grid. A one-level Oseen iterative method on a fine mesh is further studied under a weak uniqueness condition. Moreover, the stability and error estimate are rigorously carried out, which prove that the proposed methods are stable and effective. Finally, some numerical examples corroborate the effectiveness of our theoretical analysis and the proposed methods.

A locking-free nonconforming FEM for optimal control problems governed by linear elasticity equations

In this paper, we investigate a locking-free nonconforming finite element method (FEM) for linear elasticity optimal control problems (OCPs) via the variational control discretization approach. Due to the fact that the finite element interpolation operator is identical to the Ritz projection, optimal error estimates are established for the control, state and adjoint state variables, respectively. Numerical experiments are also provided to validate the theoretical results.

Convergence of adaptive nonconforming finite element method for Stokes optimal control problems

This paper aims at proving the convergence and quasi-optimality of an adaptive nonconforming finite element method for Stokes distributed control problems with pointwise control constraints. Nonconforming P1/P0 pair (Crouzeix–Raviart elements) and variational discretization are used to approximate the state equation and the control variable, respectively. A posteriori error estimates with upper and lower bounds are first derived for the state and adjoint variables. Then we prove the contraction property for the sum of the energy error of the state and adjoint state and the scaled error estimator on two consecutive adaptive meshes. The resulting linear convergence is finally used to show the quasi-optimal convergence rate of the adaptive algorithm. Additionally, some numerical results are provided to support our theoretical analysis.