Nonconforming Finite Element Approximation Research Articles

Nonconforming finite element approximations and the analysis of Nitsche's method for a singularly perturbed quad-curl problem in three dimensions

Nonconforming finite element approximations and the analysis of Nitsche's method for a singularly perturbed quad-curl problem in three dimensions

A numerical method for MHD Stokes model with applications in blood flow

Abstract The magnetohydrodynamic (MHD) Stokes equations have several applications in the field of biofluid dynamics. In the present study, we propose the staggered finite volume method (S-FVM) for MHD Stokes equations and establish its equivalence to a nonconforming finite element approximation. We also theoretically establish the convergence of the proposed S-FVM. The error estimation is carried out in an unstructured grid framework which is known for its flexibility and robustness in dealing with complex domains. The apriori estimate shows that the L 2 {L}_{2} -norm of the error for the pressure and velocity components is of order h h , the spacial grid size. After validating the numerical performance of the scheme against benchmark test cases, we do numerical simulations for the blood flow through an injured arteriole and analyze the influence of the magnetic force on hemodynamics in the arteriole under an injured condition.

Least-Squares Methods with Nonconforming Finite Elements for General Second-Order Elliptic Equations

In this paper, we study least-squares finite element methods (LSFEM) for general second-order elliptic equations with nonconforming finite element approximations. The equation may be indefinite. For the two-field potential-flux div LSFEM with Crouzeix–Raviart (CR) element approximation, we present three proofs of the discrete solvability under the condition that mesh size is small enough. One of the proof is based on the coerciveness of the original bilinear form. The other two are based on the minimal assumption of the uniqueness of the solution of the second-order elliptic equation. A counterexample shows that div least-squares functional does not have norm equivalence in the sum space of $$H^1$$ and CR finite element spaces. Thus it cannot be used as an a posteriori error estimator. Several versions of reliable and efficient error estimators are proposed for the method. We also propose a three-filed potential-flux-intensity div-curl least-squares method with general nonconforming finite element approximations. The norm equivalence in the abstract nonconforming piecewise $$H^1$$ -space is established for the three-filed formulation on the minimal assumption of the uniqueness of the solution of the second-order elliptic equation. The three-filed div-curl nonconforming formulation thus has no restriction on the mesh size, and the least-squares functional can be used as the built-in a posteriori error estimator. Under some restrictive conditions, we also discuss a potential-flux div-curl least-squares method.

Residual‐based a posteriori error estimates for nonconforming finite element approximation to parabolic interface problems

AbstractIn this paper, we derive a residual‐based a posteriori error estimates for nonconforming finite element approximation to parabolic interface problems. The present approach does not involve the Helmholtz decomposition while analyzing the reliability of the estimator. The constants involved in the estimators are independent of the jump of the diffusion coefficient across the interface, and the quasi‐monotonocity assumption on the diffusion coefficient is relaxed. The reliability bound of the estimator consists of the element residual, the edge flux jump and the edge solution jump. The efficiency of the estimator is analyzed by employing a coarsening strategy introduced by Chen and Feng's study. We derive both global upper bound and a local lower bound for the error and an adaptive space–time algorithm is prescribed using the derived estimators. Numerical results illustrating the behavior of the estimators are provided.

A Posteriori Error Estimates for a Nonconforming Finite Element Discretization of the Stokes–Biot System

This paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the problem defining the interaction between a free fluid and poroelastic structure. The free fluid is governed by the Stokes equations, while the flow in the poroelastic medium is modeled using the Biot poroelasticity system. Equilibrium and kinematic conditions are imposed on the interface. The approach utilizes the same nonconforming Crouzeix–Raviart element discretization on the entire domain. For this discretization, we derive a residual indicator based on the jumps of the normal derivative of the nonconforming approximation. Lower and upper bounds form the main results with minimal assumptions on the mesh.

Superconvergence of nonconforming finite element approximation for the second order elliptic problem

In this paper a general superconvergence of nonconforming finite element method for the second order elliptic problem is derived. In order to verify and support the theoretical results numerical examples are given.

Morley finite element method for the von Kármán obstacle problem

This paper focusses on the von Kármán equations for the moderately large deformation of a very thin plate with the convex obstacle constraint leading to a coupled system of semilinear fourth-order obstacle problem and motivates its nonconforming Morley finite element approximation. The first part establishes the well-posedness of the von Kármán obstacle problem and also discusses the uniqueness of the solution under an a priori and an a posteriori smallness condition on the data. The second part of the article discusses the regularity result of Frehse from 1971 and combines it with the regularity of the solution on a polygonal domain. The third part of the article shows an a priori error estimate for optimal convergence rates for the Morley finite element approximation to the von Kármán obstacle problem for small data. The article concludes with numerical results that illustrates the requirement of smallness assumption on the data for optimal convergence rate.

Generalized Prager–Synge identity and robust equilibrated error estimators for discontinuous elements

The well-known Prager–Synge identity is valid in H1(Ω) and serves as a foundation for developing equilibrated a posteriori error estimators for continuous elements. In this paper, we introduce a new identity, that may be regarded as a generalization of the Prager–Synge identity, to be valid for piecewiseH1(Ω) functions for diffusion problems.For nonconforming finite element approximation of arbitrary odd order, we improve the current methods by proposing a fully explicit approach that recovers an equilibrated flux in H(div;Ø) through a local element-wise scheme. The local efficiency for the recovered flux is robust with respect to the diffusion coefficient jump regardless of its distribution.For discontinuous elements, we note that the typical approach of recovering a H1 function for the nonconforming error can be proved robust only under some restrictive assumptions. To promote the unconditional robustness of the error estimator with respect to the diffusion coefficient jump, we propose to recover a gradient in H(curl;Ω) space through a simple explicit averaging technique over facets. Our resulting error estimator is proved to be globally reliable and locally efficient regardless of the coefficient distribution. Nevertheless, the reliability constant is no longer to be 1.

Error estimation for second‐order partial differential equations in nonvariational form

AbstractSecond‐order partial differential equations (PDEs) in nondivergence form are considered. Equations of this kind typically arise as subproblems for the solution of Hamilton–Jacobi–Bellman equations in the context of stochastic optimal control, or as the linearization of fully nonlinear second‐order PDEs. The nondivergence form in these problems is natural. If the coefficients of the diffusion matrix are not differentiable, the problem cannot be transformed into the more convenient variational form. We investigate tailored nonconforming finite element approximations of second‐order PDEs in nondivergence form, utilizing finite‐element Hessian recovery strategies to approximate second derivatives in the equation. We study both approximations with continuous and discontinuous trial functions. Of particular interest are a priori and a posteriori error estimates as well as adaptive finite element methods. In numerical experiments our method is compared with other approaches known from the literature.

On <inline-formula><tex-math id="M1">$ P_1 $</tex-math></inline-formula> nonconforming finite element aproximation for the Signorini problem

<p style='text-indent:20px;'>The main aim of this paper is to study the <inline-formula><tex-math id="M2">$ P_1 $</tex-math></inline-formula> nonconforming finite element approximations of the variational inequality arisen from the Signorini problem. We describe the finite dimensional closed convex cone approximation in a meanvalue-oriented sense. In this way, the optimal convergence rate <inline-formula><tex-math id="M3">$ O(h) $</tex-math></inline-formula> can be obtained by a refined analysis when the exact solution belongs to <inline-formula><tex-math id="M4">$ H^{2}(\Omega) $</tex-math></inline-formula> without any assumption. Furthermore, we also study the optimal convergence for the case <inline-formula><tex-math id="M5">$ u\in H^{1+\nu}(\Omega) $</tex-math></inline-formula> with <inline-formula><tex-math id="M6">$ \frac{1}{2}<\nu<1 $</tex-math></inline-formula>.</p>

Crouzeix–Raviart Finite Element Approximation for the Parabolic Obstacle Problem

Abstract We introduce and study a fully discrete nonconforming finite element approximation for a parabolic variational inequality associated with a general obstacle problem. The method comprises of the Crouzeix–Raviart finite element method for space discretization and implicit backward Euler scheme for time discretization. We derive an error estimate of optimal order 𝒪 ⁢ ( h + Δ ⁢ t ) {\mathcal{O}(h+\Delta t)} in a certain energy norm defined precisely in the article. We only assume the realistic regularity u t ∈ L 2 ⁢ ( 0 , T ; L 2 ⁢ ( Ω ) ) {u_{t}\in L^{2}(0,T;L^{2}(\Omega))} and moreover the analysis is performed without any assumptions on the speed of propagation of the free boundary. We present a numerical experiment to illustrate the theoretical order of convergence derived in the article.

Quasi-Optimality of Adaptive Mixed FEMs for Non-selfadjoint Indefinite Second-Order Linear Elliptic Problems

Abstract The well-posedness and the a priori and a posteriori error analysis of the lowest-order Raviart–Thomas mixed finite element method (MFEM) has been established for non-selfadjoint indefinite second-order linear elliptic problems recently in an article by Carstensen, Dond, Nataraj and Pani (Numer. Math., 2016). The associated adaptive mesh-refinement strategy faces the difficulty of the flux error control in H ⁢ ( div , Ω ) {H({\operatorname{div}},\Omega)} and so involves a data-approximation error ∥ f - Π 0 ⁢ f ∥ {\lVert f-\Pi_{0}f\rVert} in the L 2 {L^{2}} norm of the right-hand side f and its piecewise constant approximation Π 0 ⁢ f {\Pi_{0}f} . The separate marking strategy has recently been suggested with a split of a Dörfler marking for the remaining error estimator and an optimal data approximation strategy for the appropriate treatment of ∥ f - Π 0 ⁢ f ∥ L 2 ⁢ ( Ω ) {\|f-\Pi_{0}f\|_{L^{2}(\Omega)}} . The resulting strategy presented in this paper utilizes the abstract algorithm and convergence analysis of Carstensen and Rabus (SINUM, 2017) and generalizes it to general second-order elliptic linear PDEs. The argument for the treatment of the piecewise constant displacement approximation u RT {u_{{\mathrm{RT}}}} is its supercloseness to the piecewise constant approximation Π 0 ⁢ u {\Pi_{0}u} of the exact displacement u. The overall convergence analysis then indeed follows the axioms of adaptivity for separate marking. Some results on mixed and nonconforming finite element approximations on the multiply connected polygonal 2D Lipschitz domain are of general interest.

On Semicoercive Variational-Hemivariational Inequalities—Existence, Approximation, and Regularization

In this paper, we are concerned with semicoercive variational-hemivariational inequalities that encompass nonlinear semicoercive monotone variational inequalities (VIs) and pseudomonotone VIs in reflexive Banach spaces and hemivariational inequalities (HVIs) in function spaces. We present existence, approximation, and regularization results. Our approach to our existence result is based on recession arguments. We employ regularization techniques of nondifferentiable optimization to smooth the jumps in the hemivariational term. We treat nonconforming finite element approximation via Mosco convergence. As an example, we consider a semicoercive unilateral boundary value problem with nonmonotone boundary conditions that models a unilateral contact problem for a nonlinear elastic body under a nonmonotone friction law.

A Nonconforming Finite Element Method for an Acoustic Fluid-Structure Interaction Problem

Abstract We study a nonconforming finite element approximation of the vibration modes of an acoustic fluid-structure interaction. Displacement variables are used for both the fluid and the solid. The numerical scheme is based on an irrotational fluid displacement formulation and hence it is free of spurious eigenmodes. The method uses weakly continuous P 1 {P_{1}} vector fields for the fluid and classical piecewise linear elements for the solid, and it has O ⁢ ( h 2 ) {O(h^{2})} convergence for the eigenvalues on properly graded meshes. The theoretical results are confirmed by numerical experiments.

A posteriori error analysis of nonconforming finite element methods for convection–diffusion problems

A unified framework is established for the a posteriori error analysis of nonconforming finite element approximations to convection–diffusion problems. Under some certain conditions, the theory assures the semi-robustness of residual error estimates in the usual energy norm and the robustness in a modified norm, and applies to several nonconforming finite elements, such as the Crouzeix–Raviart triangular element, the nonconforming rotated (NR) parallelogram element of Rannacher and Turek, the constrained NR parallelogram element, etc. Based on the general error decomposition in different norms, we show that the key ingredients of error estimation are the existence of a bounded linear operator Π:Vhc→Vhnc with some elementary properties and the estimation on the consistency error related to the particular numerical scheme. The numerical results are presented to illustrate the practical behavior of the error estimator and check the theoretical predictions.

Extension of NXFEM to nonconforming finite elements

In this paper, we consider triangular nonconforming finite element approximations of an interface elliptic problem. We propose two extensions of the conforming Nitsche’s extended finite element method to the nonconforming case. The first one is obtained by adding stabilisation terms on the cut edges, and the second one by modifying the Crouzeix–Raviart basis functions on the cut cells. Both discrete problems are uniformly stable and yield optimal a priori error estimates, uniformly with respect to the diffusion parameters. Moreover, we show that they exhibit the same robustness with respect to the position of the interface as the classical conforming method. We then validate these results numerically. Finally, we propose a nonconforming approximation of the interface Stokes problem based on the modified Crouzeix–Raviart elements and we illustrate it numerically.

Analytical solution and nonconforming finite element approximation for the 2D multi-term fractional subdiffusion equation

The current paper focuses on analytical solution and numerical approximation for the two-dimensional(2D) multi-term time fractional subdiffusion equation. In order to derive an effective numerical solution of the equation, an unconditionally stable fully-discrete approximate scheme is established by using a modified L1 approximation and spatial nonconforming finite element method, which has many advantages comparing with conforming one. Moreover, by employing the Crouzeix–Raviart type EQ1rot nonconforming element and dealing with fractional derivatives skillfully, temporal optimal order error estimates and spatial optimal convergence rates in both L2-norm and broken energy norm are proposed without restrictions between time step and mesh size. Then, several numerical results have been provided to give an insight into the efficiency and reliability of the theoretical analysis. Finally, the derivation process of the analytical solution is presented by means of a method of separating variables.

Non-conforming finite element methods for transmission eigenvalue problem

The transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering theory. In this paper, for the Helmholtz transmission eigenvalue problem, we give a weak formulation which is a nonselfadjoint linear eigenvalue problem. Based on the weak formulation, we first discuss the non-conforming finite element approximation, and prove the error estimates of the discrete eigenvalues obtained by the Adini element, the Morley–Zienkiewicz element, the modified-Zienkiewicz element et al. And we report some numerical examples to validate the efficiency of our approach for solving transmission eigenvalue problem.

Numerical Experiments Using MATLAB: Superconvergence of Nonconforming Finite Element Approximation for Second-Order Elliptic Problems

The superconvergence in the finite element method is a phenomenon in which the fi-nite element approximation converges to the exact solution at a rate higher than the optimal order error estimate. Wang proposed and analyzed superconvergence of the conforming finite element method by L2-projections. However, since the conforming finite element method (CFEM) requires a strong continuity, it is not easy to construct such finite elements for the complex partial differential equations. Thus, the nonconforming finite element method (NCFEM) is more appealing computationally due to better stability and flexibility properties compared to CFEM. The objective of this paper is to establish a general superconvergence result for the nonconforming finite element approximations for second-order elliptic problems by L2-projection methods by applying the idea presented in Wang. MATLAB codes are published at https://github.com/annaleeharris/Superconvergence-NCFEM for anyone to use and to study. The results of numerical experiments show great promise for the robustness, reliability, flexibility and accuracy of superconvergence in NCFEM by L2- projections.

Guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem

We derive guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem. First, an abstract a posteriori error bound is derived under a special equilibration condition. Based on conservative flux reconstruction, two error estimators are proposed and provide actual upper error bounds in the usual energy norm without unknown constants, one of which can be directly constructed without solving local Neumann problems and provide practical computable error bounds. The error estimators also provide local lower bounds but with the multiplicative constants dependent on the diffusion coefficient and mesh size, where the constants can be bounded for enough small mesh size comparable with the square root of the diffusion coefficient. By adding edge jumps with weights to the energy norm, two modified error estimators with additional edge tangential jumps are shown to be robust with respect to the diffusion coefficient and provide guaranteed upper bounds on the error in the modified norm. Finally, the performance of the estimators are illustrated by the numerical results.