Nonconforming Finite Element Research Articles

A mixed-dimensional CutFEM methodology for the simulation of fibre-reinforced composites

We develop a novel unfitted finite element solver for composite materials with quasi-1D fibrous reinforcements. The method belongs to the class of mixed-dimensional non-conforming finite element solvers. The fibres are treated as 1D structural elements that may intersect the mesh of the embedding structure in an arbitrary manner. No meshing of the unidimensional elements is required. Instead, fibre solution fields are described using the trace of the background mesh. A regularised “cut” finite element formulation is carefully designed to ensure that analyses using such non-conforming finite element descriptions are stable. We also design a dedicated primal/dual operator splitting scheme to resolve the coupling between structure and fibrous reinforcements efficiently. The novel computational strategy is applied to the solution of stiff computational models whereby fibrous reinforcements may lose their bond to the embedding material above a certain level of stress. It is shown that the primal-dual 1D/3D CutFEM scheme is convergent and well-behaved in variety of scenarios involving such highly nonlinear structural computations.

Nonconforming finite element method for a generalized nonlinear Schrödinger equation

In this paper, an efficient nonconforming finite element method (FEM) is studied with EQ1rot element for a generalized nonlinear Schrödinger equation. First, we prove a novel result of the consistency error estimate with order O(h2) for EQ1rot element which leads to the superconvergent error estimate in broken H1-norm for semi-discrete scheme, while previous work only derive convergent results for this element. Second, a linearized backward Euler scheme is established and a time-discrete system is introduced to split the error into two parts, the temporal error and the spatial error. By using a rigorous analysis for the regularity of the time-discrete system and the proved characters of EQ1rot element, the optimal L2-error estimate is obtained without any time-step restrictions, which leads to the numerical solution can be bounded in L∞-norm by an inverse inequality unconditionally. Then, the supercloseness estimate is arrived at with the above achievements. Third, global superconvergence results are deduced through interpolated postprocessing technique. At last, numerical examples are provided to confirm the theoretical analysis. Here, h is the subdivision parameter, and τ is the time step.

Nonconforming Finite Element Methods for the Constrained Optimal Control Problems Governed by Nonsmooth Elliptic Equations

In this paper, nonconforming finite element methods (FEMs) are proposed for the constrained optimal control problems (OCPs) governed by the nonsmooth elliptic equations, in which the popular $$EQ_1^{rot}$$ element is employed to approximate the state and adjoint state, and the piecewise constant element is used to approximate the control. Firstly, the convergence and superconvergence properties for the nonsmooth elliptic equation are obtained by introducing an auxiliary problem. Secondly, the goal-oriented error estimates are obtained for the objective function through establishing the negative norm error estimate. Lastly, the methods are extended to some other well-known nonconforming elements.

A Stabilized Dual Mixed Hybrid Finite Element Method with Lagrange Multipliers for Three-Dimensional Elliptic Problems with Internal Interfaces

This work studies an elliptic boundary value problem with diffusive, advective and reactive terms, in a three-dimensional domain composed of two media separated by a selective interface. For the numerical approximation of the problem we propose a novel approach that combines, for the first time: (1) a dual mixed hybrid (DMH) finite element method (FEM) based on the lowest order Raviart–Thomas space (RT0); (2) a Three-Field formulation; and (3) a Streamline Upwind/Petrov–Galerkin (SUPG) stabilization method. After proving that the weak formulation of the proposed method and its numerical counterpart are both uniquely solvable and that the finite element scheme enjoys optimal convergence properties with respect to the discretization parameter, we present an efficient implementation based on static condensation, which reduces the method to a nonconforming finite element approach on a grid made by three-dimensional simplices. Extensive computational tests indicate that: (1) the theoretical convergence properties are verified; (2) the DMH-RT0 FEM is accurate and stable even in the presence of marked interface jump discontinuities in the solution and its associated normal flux; and (3) in the case of strongly dominating advective terms, the SUPG stabilization resolves accurately steep boundary and/or interior layers without introducing spurious unphysical oscillations or excessive smearing of the solution front.

Improved L2 and H1 error estimates for the Hessian discretization method

AbstractThe Hessian discretization method (HDM) for fourth‐order linear elliptic equations provides a unified convergence analysis framework based on three properties namely coercivity, consistency, and limit‐conformity. Some examples that fit in this approach include conforming and nonconforming finite element methods (ncFEMs), finite volume methods (FVMs) and methods based on gradient recovery operators. A generic error estimate has been established in L2, H1, and H2‐like norms in literature. In this paper, we establish improved L2 and H1 error estimates in the framework of HDM and illustrate it on various schemes. Since an improved L2 estimate is not expected in general for FVM, a modified FVM is designed by changing the quadrature of the source term and a superconvergence result is proved for this modified FVM. In addition to the Adini ncFEM, in this paper, we show that the Morley ncFEM is an example of HDM. Numerical results that justify the theoretical results are also presented.

Surface Crouzeix–Raviart element for the Laplace–Beltrami equation

This paper is concerned with the nonconforming finite element discretization of geometric partial differential equations. In specific, we construct a surface Crouzeix-Raviart element on the linear approximated surface, analogous to a flat surface. The optimal error estimations are established even though the presentation of the geometric error. By taking the intrinsic viewpoint of manifolds, we introduce a new superconvergent gradient recovery method for the surface Crouzeix-Raviart element using only the information of discretization surface. The potential of serving as an asymptotically exact {\it a posteriori} error estimator is also exploited. A series of benchmark numerical examples are presented to validate the theoretical results and numerically demonstrate the superconvergence of the gradient recovery method.

Global superconvergence analysis of a nonconforming FEM for Neumann boundary OCPs with elliptic equations

In this paper, a nonconforming finite element method (FEM) is proposed for the Neumann type boundary optimal control problems (OCPs) governed by elliptic equations. The state and adjoint state are approximated by the nonconforming elements, and the control is approximated by the orthogonal projection through the adjoint state. Some superclose behaviors are derived by full use of the distinguish characters of this element, such as the interpolation operator equals to the Ritz projection, and the consistency error is higher than its interpolation error in the broken energy norm. After that, the global superconvergence results are obtained by employing the so-called post-interpolation technique. Finally, some numerical results are provided to verify the theoretical analysis.

How to Prove the Discrete Reliability for Nonconforming Finite Element Methods

Optimal convergence rates of adaptive finite element methods are well understood in terms of the axioms of adaptivity. One key ingredient is the discrete reliability of a residual-based a posteriori error estimator, which controls the error of two discrete finite element solutions based on two nested triangulations. In the error analysis of nonconforming finite element methods, like the Crouzeix-Raviart or Morley finite element schemes, the difference of the piecewise derivatives of discontinuous approximations to the distributional gradients of global Sobolev functions plays a dominant role and is the object of this paper. The nonconforming interpolation operator, which comes natural with the definition of the aforementioned nonconforming finite element in the sense of Ciarlet, allows for stability and approximation properties that enable direct proofs of the reliability for the residual that monitors the equilibrium condition. The novel approach of this paper is the suggestion of a right-inverse of this interpolation operator in conforming piecewise polynomials to design a nonconforming approximation of a given coarse-grid approximation on a refined triangulation. The results of this paper allow for simple proofs of the discrete reliability in any space dimension and multiply connected domains on general shape-regular triangulations beyond newest-vertex bisection of simplices. Particular attention is on optimal constants in some standard discrete estimates listed in the appendices.

A New Numerical Method for an Asymptotic Coupled Model of Fractured Media Aquifer System

An asymptotic model coupling three-dimensional and two-dimensional equations is considered to demonstrate the flow in fractured media aquifer system in this paper. The flow is governed by Darcy’s law both in fractures and surrounding porous media. A new anisotropic and nonconforming finite element is constructed to solve the three-dimensional Darcy equation. The existence and uniqueness of the coupled solutions are deduced. Optimal error estimates are obtained in $$L^2$$ and $$H^1$$ norms. Numerical experiments show the accuracy and efficiency of the presented method. With the same number of nodal points and the same amount of computational costs, the results obtained by using the new element are much better than those by both $$Q_{1}$$ conforming element and Wilson nonconforming element on the same meshes.

A Nonconforming Nitsche's Extended Finite Element Method for Elliptic Interface Problems

A Nonconforming Nitsche's Extended Finite Element Method for Elliptic Interface Problems

An Error Analysis Method SPP-BEAM and a Construction Guideline of Nonconforming Finite Elements for Fourth Order Elliptic Problems

An Error Analysis Method SPP-BEAM and a Construction Guideline of Nonconforming Finite Elements for Fourth Order Elliptic Problems

A Hybridized Crouziex-Raviart Nonconforming Finite Element and Discontinuous Galerkin Method for a Two-Phase Flow in the Porous Media

A Hybridized Crouziex-Raviart Nonconforming Finite Element and Discontinuous Galerkin Method for a Two-Phase Flow in the Porous Media

On the Uniform Convergence of the Weak Galerkin Finite Element Method for a Singularly-Perturbed Biharmonic Equation

For the biharmonic equation or this singularly-perturbed biharmonic equation, lower order nonconforming finite elements are usually used. It is difficult to construct high order $$C^1$$ conforming, or nonconforming elements, especially in 3D. A family of any quadratic or higher order weak Galerkin finite elements is constructed on 2D polygonal grids and 3D polyhedral grids for solving the singularly-perturbed biharmonic equation. The optimal order of convergence, up to any order the smooth solution can have, is proved for this method, in a discrete $$H^2$$ norm. Under a full elliptic regularity $$H^4$$ assumption, the $$L^2$$ convergence achieves the optimal order as well, in 2D and 3D. Numerical tests are presented verifying the theory.

A Mixed Formulation of Stabilized Nonconforming Finite Element Method for Linear Elasticity

A Mixed Formulation of Stabilized Nonconforming Finite Element Method for Linear Elasticity

The adaptive finite element method for the P-Laplace problem

We consider the adaptive finite element method (AFEM) for the P-Laplace problem, −div(|∇u|p−2∇u)=f. A posteriori and priori error analysis of conforming and nonconforming finite element method are measured in the new framework. A number of experiments confirm the effective decay rates of the AFEM.

Finite element solution to 3-D airborne time-domain electromagnetic problems in complex geological media using non-conforming hexahedral meshes

We propose the approach to 3-D modeling of the electromagnetic (EM) field in time-domain airborne EM problems in complex media including rugged topography and curved borders between the geological layers, target objects of different shapes, lateral inhomogeneity within subhorizontal layers. For discretization of the electromagnetic field in space, the vector finite element method is used. We propose the approach to generating the optimized non-conforming finite element meshes with hexahedral cells in which several finite elements can be attached to the face of the other finite element which allows achieving the significant reduction of computational costs in comparison with the regular meshes on which the numerical solution has the same accuracy. The optimized non-conforming finite element mesh in a complex medium with the topography and curved borders of layers is generated with the use of special transformations. The method of constructing the finite element approximations on these non-conforming hexahedral meshes is proposed. For discretization of the electromagnetic field in time, the implicit three-point scheme “with backward overstepping” is proposed. For solving the finite element SLAE, we use the direct solver PARDISO the effectiveness of which is achieved due to the use of time stepping with a piecewise constant step and grouping the calculations of the electromagnetic field in time steps and transmitter positions. In order to increase the accuracy of calculating the signal in the receiver, a special procedure of smoothing the magnetic field B in space and special scheme of approximating the derivative of B in time are proposed. The verification of the proposed approaches is performed using the solution for the 1-D model and solutions for 3-D models obtained by other authors. We present the results of the comparative analysis of different time schemes in finite element calculating the electromagnetic field and in calculating the signal (in the form of EMF) in the receiver and choose the optimal step and the number of subintervals with the constant steps into which the entire modeling time interval is divided. We perform the comparative analysis of stair-like approximation of curved surfaces with the use of rectangular parallelepipeds and approximation with the use of the hexahedral elements and choose the optimal parameters of generating the finite element meshes. We present the results of the analysis of the possibility of replacing the model with topography with the “flat” 3-D model in which the topography is taken into account by correcting the airborne system height. For the “flat” 3-D model, we present the results of the analysis of the computational effectiveness through the use of the primary-secondary field approach with automatic choice of a horizontally-layered medium for calculating the primary field. For the complex geolelectrical model with topography, laterally-inhomogeneous overburden with curved boundaries, target objects under overburden and for a great number of airborne system positions, the computational costs are presented for sequential and parallel modes for modeling time from 0.01 ms to 1 ms and 10 ms. We present the analysis of the topography effect, the influence of the inhomogeneities in the overburden and target objects in the complex media and show that in this conditions, to discover the target objects using the form of the signal is practically impossible, i.e. in order to discover the target objects in these complex media, it is required to perform a very fine 3-D interpretation for carrying out of which fast and high accuracy forward 3-D modeling is of great importance. We obtain that for calculating the field for one transmitter and the entire modeling time interval in the sequential mode on one core of PC with processor Intel i7-8700K CPU of 3.7 GHz and 32 GB of DRAM, the required computational time is about 4–6 s. The required computational time in the parallel mode with the use of six cores of this PC is about 0.65–1 s.

A new analysis for the convergence of the gradient discretization method for multidimensional time fractional diffusion and diffusion-wave equations

We consider as discretization in space, the GDM (Gradient Discretization Method) developed recently in Droniou et al. (2018), to approximate multidimensional time fractional diffusion and diffusion-wave equations where the fractional order α of the time derivative is respectively satisfying 0<α<1 and 1<α<2. The time fractional derivative is given in the Caputo sense. The time discretization is performed using a uniform mesh. For the time fractional diffusion equation, we derive an implicit scheme. In addition to an L∞(L2)-a priori estimate, which can be derived from a reasoning in Bradji (2018), we present and prove a new L2(H01)-a priori estimate for the discrete problem. Under the assumption that the exact solution is sufficiently smooth, these a priori estimates allow to prove error estimates in discrete norms of L2(H01) and L∞(L2). The convergence order in these estimates is optimal in the sense of two points of view. The first one is that the order in space is the same one proved in Droniou et al. (2018) for elliptic equation, whereas the second point of view is that the particular case of this order when α=1 is the same one obtained in Droniou et al. (2018) for the standard heat equation. For the time fractional diffusion-wave equation, we derive two implicit schemes. A full convergence analysis is carried out for both schemes. In particular, we develop new a priori estimates which yield error estimates in several discrete norms for each scheme. The convergence is proved to be unconditional and optimal. The convergence results are obtained thanks to a comparison with appropriately chosen auxiliary gradient schemes and to the stated a priori estimates. These results improve the ones of Bradji (2017) in which a conditional convergence is proved for a SUSHI (Scheme using Stabilization and Hybrid Interfaces) approximating a time-fractional diffusion-wave equation. For both cases of time fractional diffusion and diffusion-wave equations, we show the well-posedness in the sense that the discrete solutions depend continuously on the data of the considered problems. The stated results can be extended to multi-term time-fractional diffusion and diffusion-wave equations. One of the main features when using GDM is that their results hold for all the schemes within the framework of GDM: conforming and nonconforming finite element, mixed finite element, hybrid mixed mimetic family, some Multi-Point Flux approximation finite volume schemes, and some discontinuous Galerkin schemes. Some examples of schemes recovered by the GSs (Gradient Schemes) presented in this work are sketched. Among these examples, we quote the one presented in Jin et al. (2015) to approximate time fractional diffusion equations. This work extends and improves some results presented in brief or stated without proof in the notes Bradji (2018). We present some numerical tests using SUSHI introduced in Eymard et al. (2010) to support the theoretical results.

A type of adaptive [formula omitted] non-conforming finite element method for the Helmholtz transmission eigenvalue problem

The transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering. In this paper, for the Helmholtz transmission eigenvalue problem in Rd(d=2,3), we discuss the a posteriori error estimates and adaptive algorithm of a type of C0 non-conforming finite elements including the Morley–Zienkiewicz element and the modified-Zienkiewicz element. We give the a posteriori error estimators for approximate eigenpair, and prove their reliability and efficiency for eigenfunctions. We also analyze their reliability for eigenvalues. Numerical results not only in R2 but also in R3 indicate that the a posteriori error estimators are sharp and the adaptive algorithm is efficient for computing real and complex transmission eigenvalues as expected.

The Gradient Discretisation Method for Linear Advection Problems

Abstract We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence analysis of various numerical schemes, corresponding to the methods known to be GDMs, such as finite elements (conforming or non-conforming, standard or mass-lumped), finite volumes on rectangular or simplicial grids, and other recent methods developed for general polytopal meshes. The scheme is of centred type, with added linear or non-linear numerical diffusion. We complement the convergence analysis with numerical tests based on the mass-lumped ℙ 1 {\mathbb{P}_{1}} conforming and non-conforming finite element and on the hybrid finite volume method.

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.